Number Theory

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Recent submissions

Any replacements are listed further down

[1418] viXra:1702.0300 [pdf] submitted on 2017-02-23 13:14:42

The Goldbach Conjecture - An Emergence Effect of the Primes

Authors: Ralf Wüsthofen
Comments: 11 Pages. Older versions on http://vixra.org/abs/1403.0083

The present paper shows that a principle known as emergence lies beneath the strong Goldbach conjecture. Whereas the traditional approaches focus on the control over the distribution of the primes by means of circle method and sieve theory, we give a proof of the conjecture that is based on the constructive properties of the prime numbers, reflecting their multiplicative character within the natural numbers. With an equivalent but more convenient form of the conjecture in mind, we create a structure on the natural numbers. That structure leads to arithmetic identities which immediately imply the conjecture, more precisely, an even strengthened form of it. Moreover, we can achieve further results by generalizing the structuring. Thus, it turns out that the statement of the strong Goldbach conjecture is the special case of a general principle.
Category: Number Theory

[1417] viXra:1702.0299 [pdf] submitted on 2017-02-23 14:27:23

Elementary Proof that an Infinite Number of Factorial Primes Exist

Authors: Stephen Marshall
Comments: 7 Pages.

This paper presents a complete proof of the Factorial Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m: m = (p-1)!( 1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d) We use this proof for d = n(n!) to prove the infinitude of Factorial prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Factorial Prime possible.
Category: Number Theory

[1416] viXra:1702.0286 [pdf] submitted on 2017-02-22 16:09:30

Elementary Proof Grimm's Conjecture

Authors: Stephen Marshall
Comments: 3 Pages.

In mathematics, and in particular number theory, Grimm's Conjecture (named after Karl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128. The Formal statement defining Grimm’s Conjecture, still unproved, is as follows: Suppose n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 ≤ i ≤ k.
Category: Number Theory

[1415] viXra:1702.0285 [pdf] submitted on 2017-02-22 16:11:33

Elementary Proof that Hall’s Conjecture is False

Authors: Stephen Marshall
Comments: 3 Pages.

In mathematics, Hall's conjecture is an open question, as of 2015, on the differences cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C such that for any integers x and y for which y2 ≠ x3,
Category: Number Theory

[1414] viXra:1702.0273 [pdf] submitted on 2017-02-21 14:19:03

A Sequence of Cauchy Sequences Convergent to Almost All the Riemann Zeta Zeros

Authors: Stephen Crowley
Comments: 5 Pages. The method described can be extended so that it converges to *all* the zeros. I am just posting this preliminary version in case I get hit with an asteroid before I finish writing it up.

A sequence of Cauchy sequences which converge to (almost all) the Riemann zeros is constructed.
Category: Number Theory

[1413] viXra:1702.0271 [pdf] submitted on 2017-02-21 16:20:17

Elementary Proof that an Infinite Number of Cullen Primes Exist

Authors: Stephen Marshall
Comments: 7 Pages.

This paper presents a complete proof of the Cullen Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m: See paper for this equation, as the text in this abstract does not support the mathematical format for this equation. We use this proof for d = P2 + 1 to prove the infinitude of Cullen prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Cullen Prime Conjecture possible.
Category: Number Theory

[1412] viXra:1702.0265 [pdf] submitted on 2017-02-21 06:04:58

A New Simple Recursive Algorithm for Finding Prime Numbers Using Rosser's Theorem

Authors: Rédoane Daoudi
Comments: 7 Pages.

In our previous work (The distribution of prime numbers: overview of n.ln(n), (1) and (2)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the previous is known. We use our method with five intervals and two values for n (see Methods and results). Our preliminary results show a reduced difference between the real next prime number and the number given by our algorithm. However long-term studies are required to better estimate the next prime number and to reduce the difference when n tends to infinity. Indeed an efficient algorithm is an algorithm that could be used in practical research to find new prime numbers for instance.
Category: Number Theory

[1411] viXra:1702.0264 [pdf] submitted on 2017-02-21 01:57:28

Conjecture on a Subset of Woodall Numbers Divisible by Poulet Numbers

Authors: Marius Coman
Comments: 2 Pages.

The Woodall numbers are defined by the formula W(n) = n*2^n – 1 (see the sequence A003261 in OEIS). In this paper I conjecture that any Woodall number of the form 2^k*2^(2^k) – 1, where k ≥ 3, is either prime either divisible by a Poulet number.
Category: Number Theory

[1410] viXra:1702.0259 [pdf] submitted on 2017-02-20 10:38:56

Conjecture on a Subset of Mersenne Numbers Divisible by Poulet Numbers

Authors: Marius Coman
Comments: 3 Pages.

The Poulet numbers (or the Fermat pseudoprimes to base 2) are defined by the fact that are the only composites n for which 2^(n – 1) – 1 is divisible by n (so, of course, all Mersenne numbers 2^(n - 1) – 1 are divisible by Poulet numbers if n is a Poulet number; but these are not the numbers I consider in this paper). In a previous paper I conjectured that any composite Mersenne number of the form 2^m – 1 with odd exponent m is divisible by a 2-Poulet number but seems that the conjecture was infirmed for m = 49. In this paper I conjecture that any Mersenne number (with even exponent) 2^(p – 1) – 1 is divisible by at least a Poulet number for any p prime, p ≥ 11, p ≠ 13.
Category: Number Theory

[1409] viXra:1702.0253 [pdf] submitted on 2017-02-20 09:23:51

The Distribution of Prime Numbers: Overview of N.ln(n)

Authors: Rédoane Daoudi
Comments: 12 Pages.

The empirical formula giving the nth prime number p(n) is p(n) = n.ln(n) (from ROSSER (2)). Other studies have been performed (from DUSART for example (1)) in order to better estimate the nth prime number. Unfortunately these formulas don't work since there is a significant difference between the real nth prime number and the number given by the formulas. Here we propose a new model in which the difference is effectively reduced compared to the empirical formula. We discuss about the results and hypothesize that p(n) can be approximated with a constant defined in this work. As prime numbers are important to cryptography and other fields, a better knowledge of the distribution of prime numbers would be very useful. Further investigations are needed to understand the behavior of this constant and therefore to determine the nth prime number with a basic formula that could be used in both theoretical and practical research.
Category: Number Theory

[1408] viXra:1702.0226 [pdf] submitted on 2017-02-17 04:27:18

Probable Prime Test for Specific Class of N=k*b^n-1

Authors: Predrag Terzic
Comments: 3 Pages.

Polynomial time probable prime test for specific class of N=k*b^n-1 is introduced .
Category: Number Theory

[1407] viXra:1702.0191 [pdf] submitted on 2017-02-16 10:26:00

About the Number of Elements of Certain Real Sequences

Authors: Zeraoulia Elhadj
Comments: 8 Pages.

This note is concerned with presenting sufficient conditions to proves that the number of elements of certain real sequences is infinite.
Category: Number Theory

[1406] viXra:1702.0166 [pdf] submitted on 2017-02-14 10:18:35

Prime Density Formula2

Authors: Chongjunhuang
Comments: 10 Pages.

Prime density formula
Category: Number Theory

[1405] viXra:1702.0162 [pdf] submitted on 2017-02-14 08:01:15

Prime Density Formula

Authors: Chongjunhuang
Comments: 10 Pages.

no
Category: Number Theory

[1404] viXra:1702.0160 [pdf] submitted on 2017-02-13 16:00:14

Conjecture on the Fibonacci Numbers with an Index Equal to 2p Where P is Prime

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make the following conjecture: If F(2*p) is a Fibonacci number with an index equal to 2*p, where p is prime, p ≥ 5, then there exist a prime or a product of primes q1 and a prime or a product of primes q2 such that F(2*p) = q1*q2 having the property that q2 – 2*q1 is also a Fibonacci number with an index equal to 2^n*r, where r is prime or the unit and n natural. Also I observe that the ratio q2/q1 seems to be a constant k with values between 2.2 and 2.237; in fact, for p ≥ 17, the value of k seems to be 2.236067(...).
Category: Number Theory

[1403] viXra:1702.0157 [pdf] submitted on 2017-02-13 21:14:17

A Proof for Infinitely Many Twin Primes

Authors: Chongxi Yu
Comments: 8 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.
Category: Number Theory

[1402] viXra:1702.0150 [pdf] submitted on 2017-02-13 14:43:06

Elementary Proof of the Goldbach Conjecture

Authors: Stephen Marshall
Comments: 4 Pages.

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture: Every even integer which can be written as the sum of two primes (the strong conjecture) He then proposed a second conjecture in the margin of his letter: Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture). A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers. The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history. In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture. The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.
Category: Number Theory

[1401] viXra:1702.0136 [pdf] submitted on 2017-02-12 02:55:02

Primality Criterion for Safe Primes

Authors: Predrag Terzic
Comments: 1 Page.

Polynomial time primality test for safe primes is introduced .
Category: Number Theory

[1400] viXra:1702.0090 [pdf] submitted on 2017-02-07 08:27:42

A Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 29 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.
Category: Number Theory

[1399] viXra:1702.0030 [pdf] submitted on 2017-02-02 11:56:36

Updated Elementary Proofs of Polignac Prime Conjecture, Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture

Authors: Stephen Marshall
Comments: 8 Pages. This is an update to my proff subitted in 2014, I have simpified the submission by removing uneccessary material from the proof.

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer n: n = (p-10!(1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d) We use this proof for d = 2k to prove the infinitude of Polignac prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Polignac Prime Conjecture possible. Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.
Category: Number Theory

[1398] viXra:1702.0027 [pdf] submitted on 2017-02-02 09:19:28

A Quadruplet of Numbers

Authors: Dragan Turanyanin
Comments: 3 Pages.

Three real numbers are introduced via related infinite series. With e, together they complete a quadruplet.
Category: Number Theory

[1397] viXra:1701.0682 [pdf] submitted on 2017-01-30 17:11:35

About the Distribution of Prime Numbers

Authors: Federico Gabriel
Comments: 2 Pages.

In this article, a prime number distribution formula is given. The formula is based on the periodic property of the sine function and an important trigonometric limit.
Category: Number Theory

[1396] viXra:1701.0664 [pdf] submitted on 2017-01-29 15:23:52

(VBGC 1.1 the Conjecture Only 29.01.2017 7 Pages) the " Vertical " (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.1) as Applied on Primes with (Recursive) Prime Indexes (o-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 7 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths
Category: Number Theory

[1395] viXra:1701.0647 [pdf] submitted on 2017-01-28 03:12:53

Essai su L'Hypothèse de Riemann

Authors: M. MADANI Bouabdallah
Comments: 7 Pages. Seul M. Andrzej Schinzel (IMPAN) a accepté d'examiner mon texte début janvier,il en a résulté 3 observations.Les 2 premières ont été solutionnées (lemmes 1 et 2) et la 3ème a fait l'objet d'un désaccord.J'ai demandé l'arbitrage à MM. Pierre Deligne,E. Bom

J.P. Gram (1903)writes p.298 of his paper 'Note sur les zéros de la fonction zéta de Riemann' : 'Mais le résultat le plus intéressant qu'ait donné ce calcul consiste en ce qu'il révèle l'irrégularité qui se trouve dans la série des α. Il est très probable que ces racines sont liées intimement aux nombres premiers. La recherche de cette dépendance, c'est-à-dire la manière dont une α donnée est exprimée au moyen des nombres premiers sera l'objet d'études ultérieures.' Also the proof of the Riemann hypothesis is based on the definition of an application between the set P of the prime numbers and the set S of the zeros of ζ.
Category: Number Theory

[1394] viXra:1701.0630 [pdf] submitted on 2017-01-26 22:23:47

Hyperspheres in Fermat's Last Theorem

Authors: Kelvin Kian Loong Wong
Comments: 17 Pages. French translation for abstract and keywords

This paper provides a potential pathway to a formal simple proof of Fermat's Last Theorem. The geometrical formulations of n-dimensional hypergeometrical models in relation to Fermat's Last Theorem are presented. By imposing geometrical constraints pertaining to the spatial allowance of these hypersphere configurations, it can be shown that a violation of the constraints confirms the theorem for n equal to infinity to be true.
Category: Number Theory

[1393] viXra:1701.0618 [pdf] submitted on 2017-01-25 20:40:28

An Algorithmic Proof of the Twin Primes Conjecture and the Goldbach Conjecture

Authors: Juan G. Orozco
Comments: 8 Pages.

Abstract. This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in a Euler product\prod{1-\frac{a}{p}}.  
Category: Number Theory

[1392] viXra:1701.0602 [pdf] submitted on 2017-01-24 00:00:25

Conjecture on 3-Carmichael Numbers of the Form (4h+1)(4j+1)(4k+1)

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that for any 3-Carmichael number (absolute Fermat pseudoprime with three prime factors, see the sequence A087788 in OEIS) of the form (4*h + 1)*(4*j + 1)*(4*k + 1) is true that h, j and k must share a common factor (in fact, for seven from a randomly chosen set of ten consecutive, reasonably large, such numbers it is true that both j and k are multiples of h). The conjecture is probably true even for the larger set of 3-Poulet numbers (Fermat pseudoprimes to base 2 with three prime factors, see the sequence 215672 in OEIS).
Category: Number Theory

[1391] viXra:1701.0600 [pdf] submitted on 2017-01-24 02:35:20

Conjecture on 3-Carmichael Numbers of the Form (4h+3)(4j+1)(4k+3)

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that for any 3-Carmichael number (absolute Fermat pseudoprime with three prime factors, see the sequence A087788 in OEIS) of the form (4*h + 3)*(4*j + 1)*(4*k + 3) is true that (k – h) and j must share a common factor (sometimes (k – h) is a multiple of j). The conjecture is probably true even for the larger set of 3-Poulet numbers (Fermat pseudoprimes to base 2 with three prime factors, see the sequence 215672 in OEIS).
Category: Number Theory

[1390] viXra:1701.0588 [pdf] submitted on 2017-01-25 02:44:01

(VBGC 1.0 25.01.2017 15 Pages) the " Vertical " (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.0) as Applied on Primes with (Recursive) Prime Indexes (o-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 15 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths
Category: Number Theory

[1389] viXra:1701.0585 [pdf] submitted on 2017-01-23 13:26:30

Conjecture on 2-Poulet Numbers of the Form (4h+1)(4k+1)

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that for any 2-Poulet number (Fermat pseudoprime to base 2 with two prime factors, see the sequence A214305 in OEIS) of the form (4*h + 1)*(4*k + 1) is true that h and k can not be relatively primes (in fact, for sixteen from the first twenty 2-Poulet numbers of this form is true that k is a multiple of h and this is also the case for four from a randomly chosen set of five consecutive, much larger, such numbers).
Category: Number Theory

[1388] viXra:1701.0483 [pdf] submitted on 2017-01-13 13:46:54

About of Some Variants of the "Pythagorean" Decisions and Other Higher-Order Equations (Elementary Aspect)

Authors: Reuven Tint
Comments: 4 Pages. original papper in russian

Annotation. Are given in Section 1 the theorem and its proof, complementing the classical formulation of the ABC conjecture, and in Chapter 2 addressed the issue of communication with the elliptic curve Frey's "Great" Fermat's theorem.
Category: Number Theory

[1387] viXra:1701.0482 [pdf] submitted on 2017-01-13 09:00:42

Résolution de la Conjecture de Meissel-Lehmer

Authors: guilhem CICOLELLA
Comments: 4 Pages.

the only consecutives powers being 8 and 9 the probleme consisted in demonstrating that the quantities of primes numbers inferior to one billion depended on one single equation based on two different methods of calculation with congruent results,the ultimate purpose being to prove the existence of an algorithm capable of determining two intricate values more quickly than with computer(rapid mathematical system r.m.S)
Category: Number Theory

[1386] viXra:1701.0478 [pdf] submitted on 2017-01-12 13:25:43

Tom's Query: Perfect N-bics?

Authors: Tom Masterson
Comments: 1 Page. © 1965 by Tom Masterson

A number theory query related to Fermat's last theorem in higher dimensions.
Category: Number Theory

[1385] viXra:1701.0475 [pdf] submitted on 2017-01-12 10:27:06

One Observation About Primes

Authors: Nikolay Dementev
Comments: 5 Pages.

Based on the observation of randomly chosen primes it has been conjectured that the sum of digits that form any prime number should yield either even number or another prime number. The conjecture was successfully tested for the first 100 primes.
Category: Number Theory

[1384] viXra:1701.0397 [pdf] submitted on 2017-01-10 07:35:16

Fermat's Last Theorem is Wrong

Authors: Quang Nguyen Van
Comments: 1 Page.

We have found a solution of FLT for n = 3, so that FLT is wrong. In this paper, we give a counterexample ( the solution in integer for equation x^3 + y^3 = z^3 only. It is too large ( 18 digits).
Category: Number Theory

[1383] viXra:1701.0329 [pdf] submitted on 2017-01-08 11:02:17

Conjecture on the Pairs of Consecutive Primes Having the Same Number of Digits Involving Concatenation

Authors: Marius Coman
Comments: 4 Pages.

In this paper I make the following conjecture: For any pair of consecutive primes [p1, p2], p2 > p1 > 43, p1 and p2 having the same number of digits, there exist a prime q, 5 < q < p1, such that the number n obtained concatenating (from the left to the right) q with p2, then with p1, then again with q is prime. Example: for [p1, p2] = [961748941, 961748947] there exist q = 19 such that n = 1996174894796174894119 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of consecutive primes with 9 digits are 19, 17, 107, 23, 131, 47, 83, 79, 61, 277, 163, 7, 41, 13, 181, 19, 7, 37, 29 and 23 (all twenty primes lower than 300!), the corresponding primes n obtained having 20 to 24 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.
Category: Number Theory

[1382] viXra:1701.0320 [pdf] submitted on 2017-01-07 12:05:30

Conjecture on the Pairs of Twin Primes Involving Concatenation

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make the following conjecture: For any pair of twin primes [p, p + 2], p > 5, there exist a prime q, 5 < q < p, such that the number n obtained concatenating (from the left to the right) q with p + 2, then with p, then again with q is prime. Example: for [p, p + 2] = [18408287, 18408289] there exist q = 37 such that n = 37184082891840828737 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of twins with 8 digits are 19, 7, 19, 11, 23, 23, 47, 7, 47, 17, 13, 17, 17, 37, 83, 19, 13, 13, 59 and 97 (all twenty primes lower than 100!), the corresponding primes n obtained having 20 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.
Category: Number Theory

[1381] viXra:1701.0281 [pdf] submitted on 2017-01-04 06:46:28

Proof of Goldbach Conjecture

Authors: Ryujin Choe
Comments: 1 Page.

Every even integer greater than 2 can be expressed as the sum of two primes
Category: Number Theory

[1380] viXra:1701.0014 [pdf] submitted on 2017-01-03 01:34:45

Goldbach Conjecture – A Proof

Authors: Barry Foster
Comments: 2 Pages.

This is a two page attempt using simple concepts
Category: Number Theory

[1379] viXra:1701.0012 [pdf] submitted on 2017-01-02 10:39:11

On the Goldbach Conjecture

Authors: Clive Jones
Comments: 2 Pages.

An exploration of prime-number summing grids
Category: Number Theory

[1378] viXra:1701.0008 [pdf] submitted on 2017-01-02 04:55:37

Proof of Twin Prime Conjecture

Authors: Ryujin Choe
Comments: 2 Pages.

Twin primes are infinitely many
Category: Number Theory

[1377] viXra:1612.0406 [pdf] submitted on 2016-12-30 11:14:55

Two Conjectures on the Number of Primes Obtained Concatenating to the Left with Numbers Lesser Than P a Prime P (Ii)

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that there exist an infinity of primes p = 30*h + j, where j can be 1, 7, 11, 13, 17, 19, 23 or 29, such that, concatenating to the left p with a number m, m < p, is obtained a number n having the property that the number of primes of the form 30*k + j up to n is equal to p. Example: such a number p is 67 = 30*2 + 7, because there are 67 primes of the form 30*k + 7 up to 3767 and 37 < 67. I also conjecture that there exist an infinity of primes q that don’t belong to the set above, i.e. doesn’t exist m, m < q, such that, concatenating to the left q with m, is obtained a number n having the property shown. Primes can be classified based on this criteria in two sets: primes p that have the shown property like 13, 17, 23, 31, 37, 41, 47, 59, 61, 67, 71, 73, 89, 103 (...) and primes q that don’t have it like 7, 11, 19, 29, 43, 53, 79, 83, 101 (...).
Category: Number Theory

[1376] viXra:1612.0400 [pdf] submitted on 2016-12-30 02:12:38

Conjecture on Numbers N Obtained Concatenating Two Primes Related to the Number of Primes up to N (Ii)

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that for any prime p, p > 5, there exist q prime, q > p, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n, where n is the number obtained concatenating p with q, is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for p = 17 there exist q = 23 such that there are 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.
Category: Number Theory

[1375] viXra:1612.0395 [pdf] submitted on 2016-12-29 16:06:30

Conjecture on Numbers N Obtained Concatenating Two Primes Related to the Number of Primes up to N

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that there exist an infinity of numbers n obtained concatenating two primes p and q, where p = 30*k + m1 and q = 30*h + m2, p < q, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 1723 obtained concatenating the primes p = 17 and q = 23, there exist 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.
Category: Number Theory

[1374] viXra:1612.0387 [pdf] submitted on 2016-12-28 20:35:01

Every Even Integer Greater Than Six Can be Expressed as the Sum of Two co-Prime Odd Integers Atleast One of Which is a Prime

Authors: Prashanth R. Rao
Comments: 1 Page.

In this paper we prove a simple theorem that is distantly related to the Even Goldbach conjecture and is weaker than Chen’s theorem regarding the expression of any even integer as the sum of a prime number and a semiprime number. We show that any even integer greater than six can be written as the sum of two odd integers coprime to one another and atleast one of them is a prime.
Category: Number Theory

[1373] viXra:1612.0383 [pdf] submitted on 2016-12-29 01:16:00

The Sequence of Palindromes N with Property that the Number of Primes 30k+7 and 30k+11 up to N is Equal

Authors: Marius Coman
Comments: 1 Page.

In this paper I conjecture that there exist an infinity of palindromes n for which the number of primes up to n of the form 30k + 7 is equal to the number of primes up to n of the form 30k + 11 and I found the first 40 terms of the sequence of n (I also found few larger terms, as 99599, 816618 or 1001001 up to which the number of primes from the two sets, equally for each, is 1154, 8159, respectively 9817).
Category: Number Theory

[1372] viXra:1612.0294 [pdf] submitted on 2016-12-18 23:45:17

Discussion on the Negate and the Proof of ABC Conjecture

Authors: Zhang Tianshu
Comments: 21 Pages.

The ABC conjecture is both likely of the wrong and likely of the right in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we find directly a specific equality 1+2N (2N-2)=(2N-1)2 with N≥2, then set about analyzing limits of values of ε to discuss the right and the wrong of the ABC conjecture in which case satisfying 2N-1>(Rad(1, 2N(2N-2), 2N-1))1+ε . Thereby supply readers to make with a judgment concerning a truth or a falsehood which the ABC conjecture is.
Category: Number Theory

[1371] viXra:1612.0278 [pdf] submitted on 2016-12-17 11:51:55

Beal' Conjecture: Complete Proof with Numerical Examples (New Version)

Authors: Abdelmajid Ben Hadj Salem
Comments: 52 pages. In French. Submitted to the journal Functiones et Approximatio, Commentarii Mathematici. Comments welcome.

En 1997, Andrew Beal \cite{B1} avait annonc\'e la conjecture suivante : \textit{Soient $A, B,C, m,n$, et $l$ des entiers positifs avec $m,n,l > 2$. Si $A^m + B^n = C^l$ alors $A, B,$ et $C$ ont un facteur en commun}. Nous consid\'erons le polyn\^ome $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ avec $p,q$ des entiers qui d\'ependent de $A^m,B^n$ et $C^l$. La r\'esolution de $x^3-px+q=0$ nous donne les trois racines $x_1,x_2,x_3$ comme fonctions de $p,q$ et d'un param\`etre $\theta$. Comme $A^m,B^n,-C^l$ sont les seules racines de $x^3-px+q=0$, nous discutons les conditions pour que $x_1,x_2,x_3$ soient des entiers. Quatre exemples num\'eriques sont pr\'esent\'es. \\
Category: Number Theory

[1370] viXra:1612.0262 [pdf] submitted on 2016-12-16 09:29:19

The Sequence of Repnumbers N with Property that the Number of Primes 30k+11 and 30k+13 up to N is Equal

Authors: Marius Coman
Comments: 1 Page.

In my previous paper “Conjecture involving repunits, repdigits, repnumbers and also the primes of the form 30k + 11 and 30k + 13” I conjectured that there exist an infinity of repnumbers n (repunits, repdigits and numbers obtained concatenating not the unit or a digit but a number) for which the number of primes up to n of the form 30k + 11 is equal to the number of primes up to n of the form 30k + 13 and I found the first 18 terms of the sequence of n (I also found few larger terms, as 11111, 888888 and 11111111 up to which the number of primes from the two sets, equally for each, is 167, 8816, respectively 91687). In this paper I extend the search to first 40 terms of the sequence.
Category: Number Theory

[1369] viXra:1612.0260 [pdf] submitted on 2016-12-15 16:20:52

Conjecture Involving Repunits, Repdigits, Repnumbers and Also the Primes of the Form 30k+11 and 30k+13

Authors: Marius Coman
Comments: 1 Page.

In my previous paper “Conjecture on semiprimes n = p*q related to the number of primes up to n” I was wondering if there exist a class of numbers n for which the number of primes up to n of the form 30k + 1, 30k + 7, 30k + 11, 30k + 13, 30k + 17, 30k + 19, 30k + 23 and 30k + 29 is equal in each of these eight sets. I didn’t yet find such a class, but I observed that around the repdigits, repunits and repnumbers (numbers obtained concatenating not the unit or a digit but a number) the distribution of primes in these eight sets tends to draw closer and I made a conjecture about it.
Category: Number Theory

[1368] viXra:1612.0257 [pdf] submitted on 2016-12-15 10:18:33

Conjecture on Semiprimes N=pq Related to the Number of Primes up to N

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that there exist an infinity of semiprimes n = p*q, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 91 = 7*13, there exist 3 primes of the form 30*k + 7 up to 91 (7, 37 and 67) and 3 primes of the form 30*k + 13 up to 91 (13, 43 and 73).
Category: Number Theory

[1367] viXra:1612.0253 [pdf] submitted on 2016-12-15 06:24:20

Two Conjectures on the Number of Primes Obtained Concatenating to the Left with Numbers Lesser Than P a Prime P

Authors: Marius Coman
Comments: 3 Pages.

In this paper I conjecture that: (I) for any prime p of the form 6*k + 1 there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 10)/3; (II) for any prime p of the form 6*k – 1, p ≥ 11, there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 8)/3.
Category: Number Theory

[1366] viXra:1612.0223 [pdf] submitted on 2016-12-11 17:29:09

The Suggestion that 2-Probable Primes Satisfying Even Goldbach Conjecture Are Possible

Authors: Prashanth R. Rao
Comments: 2 Pages.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.
Category: Number Theory

[1365] viXra:1612.0200 [pdf] submitted on 2016-12-11 02:20:30

Conference on the Digital World, Nantes Dec 8 2016 at Lycée Clémenceau

Authors: Simon Plouffe
Comments: 28 Pages.

A presentation is made on the numerical world of mathematics. Round table on the numerical data. Une présentation du numérique à Nantes, table ronde organisée par ADN ouest au Lycée Clémenceau
Category: Number Theory

[1364] viXra:1612.0142 [pdf] submitted on 2016-12-09 02:54:12

An Identity for Generating a Special Kind of Pythagorean Quadruples

Authors: Brian Ekanyu
Comments: 6 Pages.

This paper proves an identity for generating a special kind of Pythagorean quadruples by conjecturing that the shortest is defined by a=1,2,3,4...... and b=a+1, c=ab and d=c+1. It also shows that a+d=b+c and that the surface area to volume ratio of these Pythagorean boxes is given by 4/a where a is the length of the shortest edge(side).
Category: Number Theory

[1363] viXra:1612.0140 [pdf] submitted on 2016-12-09 03:46:53

Conjecture on Odd Semiprimes Which Are Harshad Numbers that Relates Them with 2-Poulet Numbers

Authors: Marius Coman
Comments: 2 Pages.

In a previous paper I conjectured that for any largest prime factor of a Poulet number p1 with two prime factors exists a series with infinite many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the largest prime factor of p1 (see the sequence A214305 in OEIS). In this paper I conjecture that for any least prime factor of an odd Harshad number h1 with two prime factors, not divisible by 3, exists a series with infinite many Harshad numbers h2 formed this way: h2 mod (h1 - d) = d, where d is the least prime factor of p1.
Category: Number Theory

[1362] viXra:1612.0138 [pdf] submitted on 2016-12-08 15:52:25

Two Conjectures Involving Harshad Numbers, Primes and Powers of 2

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following two conjectures: (I) For any prime p, p > 5, there exist n positive integer such that the sum of the digits of the number p*2^n is divisible by p; (II) For any prime p, p > 5, there exist an infinity of positive integers m such that the sum of the digits of the number p*2^m is prime.
Category: Number Theory

[1361] viXra:1612.0101 [pdf] submitted on 2016-12-07 11:18:19

Conjecture Involving Harshad Numbers and Sexy Primes

Authors: Marius Coman
Comments: 1 Page.

In this paper I conjecture that for any pair of sexy primes (p, p + 6) there exist a prime q = p + 6*n, where n > 1, such that the number p*(p + 6)*(p + 6*n) is a Harshad number.
Category: Number Theory

[1360] viXra:1612.0072 [pdf] submitted on 2016-12-07 05:45:46

Conjecture Involving Harshad Numbers and Primes of the Form 6k+1

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that for any prime p of the form 6*k + 1 there exist an infinity of Harshad numbers of the form p*q1*q2, where q1 and q2 are distinct primes, q1 = p + 6*m and q2 = p + 6*n.
Category: Number Theory

[1359] viXra:1612.0042 [pdf] submitted on 2016-12-03 10:57:30

Proof of Twin Prime Conjecture

Authors: Safaa Abdallah Moallim
Comments: 5 Pages.

In this paper we prove that there exist infinitely many twin prime numbers by studying n when 6n±1 are primes. By studying n we show that for every n that generates a twin prime number, there has to be m>n that generates a twin prime number too.
Category: Number Theory

[1358] viXra:1611.0410 [pdf] submitted on 2016-11-30 07:48:39

The ABC Conjecture Does Not Hold Water ( Revised Version )

Authors: Zhang Tianshu
Comments: 18 Pages.

The ABC conjecture seemingly is difficult to carry conviction in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we select and adopt a specific equality 1+2N(2N-2)=(2N-1)2 with N≥2 satisfying 2N-1>(Rad(2N-2))1+ ε. Then, proceed from the analysis of the limits of values of ε to find its certain particular values, thereby finally negate the ABC conjecture once and for all.
Category: Number Theory

[1357] viXra:1611.0390 [pdf] submitted on 2016-11-29 03:29:40

Proof of Bunyakovsky's Conjecture

Authors: Robert Deloin
Comments: 13 Pages.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate infinitely many primes. The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences. As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and the binary Goldbach conjecture. The method is also used to prove that there are infinitely many primorial and factorial primes.
Category: Number Theory

[1356] viXra:1611.0373 [pdf] submitted on 2016-11-27 08:39:53

Nonlinear Curve as Proof of Fermat’s Last Theorem: a Graphical Method

Authors: Victor Christianto
Comments: 4 Pages. This paper will be submitted to Annals of Mathematics

In this paper we will give an outline of proof of Fermat’s Last Theorem using a graphical method. Although an exact proof can be given using differential calculus, we choose to use a more intuitive graphical method.
Category: Number Theory

[1355] viXra:1611.0224 [pdf] submitted on 2016-11-14 18:05:57

Sieve of Collatz

Authors: Jonas Kaiser
Comments: 11 Pages.

The sieve of Collatz is a new algorithm to trace back the non-linear Collatz problem to a linear cross out algorithm. Until now it is unproved.
Category: Number Theory

[1354] viXra:1611.0178 [pdf] submitted on 2016-11-12 09:51:56

多与少的个数区别永远会造成二个质数的距离=2

Authors: Aaron Chau
Comments: 3 Pages.

十分幸运,本文应用的是永不改变的定律(多与少),而不再是重复那类受局限的定理。 感谢数学的美妙,因为多与少的个数区别永远会造成二个质数的距离= 2。简述,= 2。
Category: Number Theory

[1353] viXra:1611.0176 [pdf] submitted on 2016-11-12 04:58:51

Conjecture that there Exist an Infinity of Even Numbers N for Which N^2 is a Harshad-Coman Number

Authors: Marius Coman
Comments: 2 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of even numbers n for which n^2 is a Harshad-Coman number and I also make a classification in four classes of all the even numbers.
Category: Number Theory

[1352] viXra:1611.0175 [pdf] submitted on 2016-11-12 05:01:08

Conjecture that there Exist an Infinity of Odd Numbers N for Which N^2 is a Harshad-Coman Number

Authors: Marius Coman
Comments: 3 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of odd numbers n for which n^2 is a Harshad-Coman number and I also make a classification in three classes of all the odd numbers greater than 1.
Category: Number Theory

[1351] viXra:1611.0172 [pdf] submitted on 2016-11-11 15:58:33

Three Conjectures Regarding Poulet Numbers and Harshad Numbers

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following three conjectures: (I) If P is both a Poulet number and a Harshad number, than the number P – 1 is also a Harshad number; (II) If P is a Poulet number divisible by 5 under the condition that the sum of the digits of P – 1 is not divisible by 5 than P – 1 is a Harshad number; (III) There exist an infinity of Harshad numbers of the form P – 1, where P is a Poulet number.
Category: Number Theory

[1350] viXra:1611.0171 [pdf] submitted on 2016-11-11 16:00:16

Conjecture that there Exist an Infinity of Poulet Numbers Which Are Also Harshad-Coman Numbers

Authors: Marius Coman
Comments: 2 Pages.

OEIS defines the notion of Harshad numbers as the numbers n with the property that n/s(n), where s(n) is the sum of the digits of n, is integer (see the sequence A005349). In this paper I define the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer and I make the conjecture that there exist an infinity of Poulet numbers which are also Harshad-Coman numbers.
Category: Number Theory

[1349] viXra:1611.0120 [pdf] submitted on 2016-11-09 07:22:21

The Generalized Goldbach’s Conjecture : Symmetry of Prime Number

Authors: Jian Ye
Comments: 3 Pages.

Goldbach’s conjecture: symmetrical primes exists in natural numbers. the generalized Goldbach’s conjecture: symmetry of prime number in the former and tolerance coprime to arithmetic progression still exists.
Category: Number Theory

[1348] viXra:1611.0089 [pdf] submitted on 2016-11-07 11:29:42

The 3n ± p Conjecture: A Generalization of Collatz Conjecture

Authors: W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache
Comments: 10 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo Kakutani) and so on. Several various generalization of the Collatz conjecture has been carried. In this paper a new generalization of the Collatz conjecture called as the 3n ± p conjecture; where p is a prime is proposed. It functions on 3n + p and 3n - p, and for any starting number n, its sequence eventually enters a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is a special case of the 3n ± p conjecture when p is 1.
Category: Number Theory

[1347] viXra:1611.0085 [pdf] submitted on 2016-11-07 06:46:24

Research Project Primus

Authors: Predrag Terzic
Comments: 32 Pages.

Some theorems and conjectures concerning prime numbers .
Category: Number Theory

[1346] viXra:1610.0356 [pdf] submitted on 2016-10-29 14:52:21

A Simple Proof of the Collatz-Gormaund Theorem (Collatz Conjecture)

Authors: Caitherine Gormaund
Comments: 2 Pages.

In which the Collatz Conjecture is proven using fairly simple mathematics.
Category: Number Theory

[1345] viXra:1610.0349 [pdf] submitted on 2016-10-28 13:23:48

Some Famous Conjectures Relative to the Consecutive Primes

Authors: Reza Farhadian
Comments: 4 Pages.

In this paper we offer the some details and particulars about some famous conjectures in relative to consecutive primes.
Category: Number Theory

[1344] viXra:1610.0313 [pdf] submitted on 2016-10-26 05:42:56

There Really Are an Infinite Number of Twin Primes, and Other Thoughts on the Distribution of Primes.

Authors: Jared Beal
Comments: 14 Pages.

This paper describes an algorithm for finding all the prime numbers. It also describes how this pattern among primes can be used to show the ratio of primes to not primes in an infinite set of X integers. It can also be used to show that the ratio of twin primes to not twin primes in an infinite set of X integers is always going to be greater than zero.
Category: Number Theory

[1343] viXra:1610.0284 [pdf] submitted on 2016-10-24 03:05:49

Доказательство гипотезы Била – следствие свойств инвариантного тождества определенного типа (элементарный аспект)

Authors: Reuven Tint
Comments: Updates: 4.3.2 - 4.3.5.. page 7

Аннотация. Предложен вариант решения гипотезы Била с помощью прямого доказательства» Великой» теоремы Ферма элементарными методами. Новыми являются «инвариантное тождество « (ключевое слово) и полученные нами приведенные в тексте работы тождества, позволившие напрямую решить ВТФ и гипотезу Била,и ряд других. Предложены также новая формулировка теорем ( п.2.1.4.), ,доказательства для n= 1,2,3,..n>2 и x,y,z>2.
Category: Number Theory

[1342] viXra:1610.0276 [pdf] submitted on 2016-10-24 00:02:00

On a Question Concerning the Littlewood Violations

Authors: John Smith
Comments: 19 Pages.

Riemann's prime-counting function R(x) looks good for every value of x we can compute, but in the light of Littlewood's result its superiority over li(x) is illusory: Ingram (1938) pointed out that 'for special values of x (as large as we please), the one approximation will deviate as widely as the other from the true value'. This note introduces a type of prime-counting function that is always better than li(x)...
Category: Number Theory

[1341] viXra:1610.0275 [pdf] submitted on 2016-10-23 13:15:42

К вопросу о связи эллиптической кривой Фрея с «Великой» теоремой Ферма (элементарный аспект)

Authors: Reuven Tint
Comments: 2 Pages.

Аннотация. Интерес к названной в заглавии проблеме вызван следующими соображениями: 1) Возьмем, к примеру, «пифагорово» уравнение, все взаимно простые решения которого опре- деляются формулами A= a^2- b^2 и B=2ab. Но если мы выберем A≠a^2- b^2 и B≠2ab как гипо- тетически «верные» решения этого уравнения, то, наверное, можно будет доказать, что, в этом случае, «пифагорово» уравнение не существует. Но оно действительно не существует для гипотетически выбранных «верных» решений. 2) Уравнение A^N+B^N = C^N и уравнение эллиптической кривой Фрея (как будет показано ниже для предложенного варианта их решения) не совместны. 3) Поэтому, как представляется, выглядит не совсем убедительной связь между уравнением эллиптической кривой Фрея и соответствующим уравнением Ферма. 4) Приведено приложение.
Category: Number Theory

[1340] viXra:1610.0274 [pdf] submitted on 2016-10-23 13:19:39

On the Question of the Relationship of the Elliptic Curve Frey with "Great" Fermat's Theorem (Elementary Aspect).

Authors: Reuven Tint
Comments: 2 Pages.

Annotation. Interest in the title problem is caused by the following considerations: 1) Take, for example, "Pythagoras' equation, all of which are relatively prime solutions determined Delyan formulas A= a^2- b^2 and B=2ab. But if we choose A≠a^2- b^2 and B≠2ab both hypo- Tethyan "correct" solutions of this equation, then perhaps it will be possible to prove that, in this case, "Pythagoras" equation exists. But it really does not exist for the selected hypothetically "true" solutions. 2) The equation A^N+B^N = C^N and the equation of the elliptic curve Frey (as will be shown below for the proposed options to solve them) are not compatible. 3) Therefore, it seems, it does not look quite convincing relationship between the equation elliptic curve Frey Farm and the corresponding equation. 4) Supplement.
Category: Number Theory

[1339] viXra:1610.0272 [pdf] submitted on 2016-10-23 13:58:45

Proposal Demonstration of Hypothesis Riemann

Authors: Luca Nascimbene
Comments: 13 Pages.

In this paper the author continue the works [6] [11] [12] and present a proposal for a demonstration on the Riemann Hypothesis and the conjecture on the multiplicity of non-trivial zeros of the Riemann s zeta.
Category: Number Theory

[1338] viXra:1610.0253 [pdf] submitted on 2016-10-21 18:17:51

Some Evidencethat the Goldbach Conjecture Could be Proved or Proved False

Authors: Filippos Nikolaidis
Comments: 10 Pages. fil_nikolaidis@yahoo.com

The present study is an effort for giving some evidence that the goldbach conjecture is not true, by showing that not all even natural numbers greater than two can be expressed as a sum of two primes. This conclusion can be drawn by showing that prime numbers are not enough –in population- so that, when added in couples, to give all the even numbers.
Category: Number Theory

[1337] viXra:1610.0183 [pdf] submitted on 2016-10-17 05:37:47

Proof of Beal's Conjecture

Authors: Edward Szaraniec
Comments: 5 Pages.

Equation constituting the Beal conjecture is rearranged and squared, then rearranged again and raised to power 4. The result, standing as an equivalent having the same property, is emerging as a singular primitive Pythagorean equation with no solution. So, the conjecture is proved. General line of proving the Pythagorean equation is observed as a moving spirit.
Category: Number Theory

[1336] viXra:1610.0172 [pdf] submitted on 2016-10-16 05:13:25

Another Proof for FERMAT’s Last Theorem

Authors: Mugur B. Răuţ
Comments: 5 Pages.

In this paper we propose another proof for Fermat’s Last Theorem (FLT). We found a simpler approach through Pythagorean Theorem, so our demonstration would be close to the times FLT was formulated. On the other hand it seems the Pythagoras’ Theorem was the inspiration for FLT. It resulted one of the most difficult mathematical problem of all times, as it was considered. Pythagorean triples existence seems to support the claims of the previous phrase.
Category: Number Theory

[1335] viXra:1610.0106 [pdf] submitted on 2016-10-10 03:35:21

A New 3n-1 Conjecture Akin to Collatz Conjecture

Authors: W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache
Comments: 9 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani's problem (after Shizuo Kakutani) and so on. In this paper a new conjecture called as the 3n-1 conjecture which is akin to the Collatz conjecture is proposed. It functions on 3n -1, for any starting number n, its sequence eventually reaches either 1, 5 or 17. The 3n-1 conjecture is compared with the Collatz conjecture.
Category: Number Theory

[1334] viXra:1610.0099 [pdf] submitted on 2016-10-08 17:28:15

Proof of Sophie Germain Conjecture

Authors: Idriss Olivier Bado
Comments: Dans ce présent document nous donnons la preuve de la conjecture de Sophie Germain en utilisant le theoreme de densité de Chebotarev ,le principe d' inclusion d'exclusion de Moivre ,la formule de Mertens . en 13 pages nous donnons une preuve convaincante

In this paper We give Sophie Germain 's conjecture proof by using Chebotarev density theorem, principle inclusion -exclusion of Moivre, Mertens formula
Category: Number Theory

[1333] viXra:1610.0083 [pdf] submitted on 2016-10-07 06:34:33

Riemann Zeta Function and Relationship to Prime Numbers

Authors: Ricardo Gil
Comments: 2 Pages.

ζ(s)=1/(((1/(2))/log(2)))+ 1/(((1/(3))/log(3)))+ 1/(((1/(4))/log(4)))+1/(((1/(5))/log(5))) is a form of Riemann Zeta Function and it shows an approximate relationship between the Riemann Zeta Function and Prime Numbers.
Category: Number Theory

[1332] viXra:1610.0082 [pdf] submitted on 2016-10-07 06:37:51

Nth Prime Equation

Authors: Ricardo Gil
Comments: 1 Page.

The classical Distribution of Primes Equation can be modified to make an Nth Prime Equation which generates the Nth Prime.
Category: Number Theory

[1331] viXra:1610.0065 [pdf] submitted on 2016-10-05 09:48:06

Lauricella Hypergeometric Series Over Finite Fields

Authors: Bing He
Comments: 14 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. These generalize some known results of Li \emph{et al} as well as several other well-known results.
Category: Number Theory

[1330] viXra:1610.0042 [pdf] submitted on 2016-10-04 12:01:31

Sophie Germain Conjecture's Proof

Authors: Idriss Olivier Bado
Comments: Dans ce présent document nous donnons la preuve du théorème de Mertens en utilisant le théorème de densité de Chebotarev ,principle d'inclusion - exclusion de Moivre,formule de Mertens en 15 pages nous donnons une élégante preuve

In this paper we give the proof of Sophie Germain's conjecture by using Chebotarev density theorem, the principle inclusion-exclusion of Moivre, Mertens formula
Category: Number Theory

[1329] viXra:1610.0034 [pdf] submitted on 2016-10-03 19:56:15

In 1991 Fermat Last Theorem Has Been Proved (1)

Authors: Chunxuan Jiang
Comments: 6 Pages.

using complex hyperbolic function we prove Fermat last theorem
Category: Number Theory

[1328] viXra:1610.0033 [pdf] submitted on 2016-10-03 20:01:14

In 1991 Fermat Last Theorem Has Been Proved (Ii)

Authors: Chunxuan Jiang
Comments: 5 Pages.

using trogonometric function we prove Fermat last theorem
Category: Number Theory

[1327] viXra:1610.0024 [pdf] submitted on 2016-10-03 09:06:13

Proving Riemann with Gamma or Euler–Mascheroni Constant (0.5772156649015328606065120900824024310421593359399)

Authors: Ricardo Gil
Comments: 2 Pages.

(1/2 Part)>1.002 (1.002, 2.16, 4.008 & 6.012) Generate Riemann Non Trivial Zero’s Off Of Critical Line. A Riemann Non Trivial Zero off the Critical Line occurs between 1 /2 or .50 and Gamma 0.577215664901532860606512090 08240243104 215 93 359399.When (1/2 Part) = (1.002 , 2.16, 4.008 & 6.012) Riemann Non Trivial Zero’s Are Off .001 To The Rt. Of The Critical Line & When (1/2 Part)= (1 / 2) A Riemann Non Trivial Zero’s Will Be On Critical Line.
Category: Number Theory

Replacements of recent Submissions

[622] viXra:1702.0273 [pdf] replaced on 2017-02-24 19:33:35

A Sequence of Cauchy Sequences Which Converge to the Imaginary Parts of the Zeros of the Riemann Zeta Function

Authors: Stephen Crowley
Comments: 9 Pages. no b.s. this time, it should be impossible to argue with this one ;)

An iteration function which has fixed-points at the zeros of the Hardy Z function is constructed and it is shown that it is impossible for this function converge to a non-real number when started with a real number. If there were any zeros of ζ(t) with Re(t)≠1/2 they would correspond to zeros of Z(t) with Im(t)≠0 and thus the constructed interation function must be able to converge for at least one real-valued starting point to a number with non-zero imaginary part, but this is impossible because the iteration function is real-valued when its argument is real. Thus, the Riemann hypothesis is shown to be true.
Category: Number Theory

[621] viXra:1702.0273 [pdf] replaced on 2017-02-23 13:25:44

A Sequence of Cauchy Sequences Convergent to the Imaginary Parts of the Zeros of the Riemann Zeta Function

Authors: Stephen Crowley
Comments: 7 Pages.

A sequence of Cauchy sequences which converge to the Riemann zeros is constructed and related to the LeClair-França criteria for the Riemann hypothesis.
Category: Number Theory

[620] viXra:1702.0157 [pdf] replaced on 2017-02-17 19:44:31

A Proof for Infinitely Many Twin Primes

Authors: Chongxi Yu
Comments: 8 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years and many “advanced mathematics tools” are used to solve them, but they are still unsolved. Based on the fundamental theorem of arithmetic and Euclid’s proof of endless prime numbers, we have proved there are infinitely many twin primes.
Category: Number Theory

[619] viXra:1702.0136 [pdf] replaced on 2017-02-15 03:23:14

Primality Criterion for Safe Primes

Authors: Predrag Terzic
Comments: 2 Pages.

Polynomial time primality test for safe primes is introduced .
Category: Number Theory

[618] viXra:1702.0136 [pdf] replaced on 2017-02-14 00:07:47

Primality Criterion for Safe Primes

Authors: Predrag Terzic
Comments: 2 Pages.

Polynomial time primality test for safe primes is introduced .
Category: Number Theory

[617] viXra:1702.0090 [pdf] replaced on 2017-02-22 08:26:54

A Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 33 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Goldbach’s conjecture is about all numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 10 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.
Category: Number Theory

[616] viXra:1702.0090 [pdf] replaced on 2017-02-22 02:59:46

A Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 33 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Goldbach’s conjecture is about all numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, here we divided any even numbers into 10 groups and primes into 4 groups, Goldbach’s conjecture will be much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless.
Category: Number Theory

[615] viXra:1702.0090 [pdf] replaced on 2017-02-13 23:18:35

A Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 24 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless. Key words: Goldbach's conjecture , fundamental theorem of arithmetic, Euclid's proof of infinite primes
Category: Number Theory

[614] viXra:1702.0090 [pdf] replaced on 2017-02-12 00:48:46

A Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 21 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. We give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic and Euclid's proof that the set of prime numbers is endless. Key words: Goldbach's conjecture , fundamental theorem of arithmetic, Euclid's proof of infinite primes
Category: Number Theory

[613] viXra:1702.0027 [pdf] replaced on 2017-02-09 15:34:07

A Quadruplet of Numbers

Authors: Dragan Turanyanin
Comments: 3 Pages.

Three real numbers are introduced via related infinite series. With e, together they complete a quadruplet.
Category: Number Theory

[612] viXra:1701.0664 [pdf] replaced on 2017-02-02 03:48:33

(VBGC 1.2c the Conjecture Only 2.02.2017 10 Pages) the "Vertical" (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.2) as Applied “iterative” Primes with (Recursive) Prime Indexes (i-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 10 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_i,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_i,p,n, with iteration order i ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_i,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with iteration order i≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (iPx is the x-th o-primeth, with iteration order i ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general iteration order i ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the i-primeths (with iteration order i≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on i-primeths
Category: Number Theory

[611] viXra:1701.0664 [pdf] replaced on 2017-01-31 05:14:05

(VBGC 1.2 - the Conjecture Only - 31.01.2017 10 Pages) the " Vertical " (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.2) as Applied on Primes with (Recursive) Prime Indexes (o-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 10 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths
Category: Number Theory

[610] viXra:1701.0618 [pdf] replaced on 2017-01-26 21:10:42

An Algorithmic Proof of the Twin Primes Conjecture and the Goldbach Conjecture

Authors: Juan G. Orozco
Comments: 9 Pages. Image of algorithm implementation example added.

This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in a Euler product\prod{1-\frac{a}{p}}.
Category: Number Theory

[609] viXra:1701.0588 [pdf] replaced on 2017-02-02 03:50:49

(VBGC 1.2c 2.02.2017 21 Pages) the "Vertical" (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.2) as Applied on “iterative” Primes with (Recursive) Prime Indexes (i-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 21 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_i,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_i,p,n, with iteration order i ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_i,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with iteration order i≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (iPx is the x-th o-primeth, with iteration order i ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general iteration order i ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the i-primeths (with iteration order i≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on i-primeths
Category: Number Theory

[608] viXra:1701.0588 [pdf] replaced on 2017-01-31 05:09:49

(VBGC 1.2 - 31.01.2017 - 19 Pages) the " Vertical " (Generalization Of) the Binary Goldbach's Conjecture (VBGC 1.2) as Applied on Primes with (Recursive) Prime Indexes (o-Primeths)

Authors: Andrei Lucian Dragoi
Comments: 19 Pages.

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_o,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_o,p,n, with order o ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_o,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with order o≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (oPx is the x-th o-primeth, with order o ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general order o ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the o-primeths (with order o≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on o-primeths
Category: Number Theory

[607] viXra:1701.0014 [pdf] replaced on 2017-02-06 00:15:30

Goldbach Conjecture – A Proof (?)

Authors: Barry Foster
Comments: 2 Pages.

This is a two page attempt using simple concepts
Category: Number Theory

[606] viXra:1701.0014 [pdf] replaced on 2017-02-02 06:02:17

Goldbach Conjecture – A Proof

Authors: Barry Foster
Comments: 2 Pages.

This is a two page attempt using simple concepts
Category: Number Theory

[605] viXra:1701.0014 [pdf] replaced on 2017-01-12 06:19:40

Goldbach Conjecture – A Proof

Authors: Barry Foster
Comments: 2 Pages.

This is a two page attempt using simple concepts
Category: Number Theory

[604] viXra:1612.0296 [pdf] replaced on 2016-12-24 13:07:51

Proof of the Riemann Hypothesis

Authors: Armando M. Evangelista Jr.
Comments: 4 Pages. typographical error on the abstract

ABSTRACT Riemann Hypothesis states that all the non-trivial zeros of the zeta function ζ(s) have real part equal to 1⁄2. It is the purpose of this present work to prove that the Riemann Hypothesis is true.
Category: Number Theory

[603] viXra:1612.0296 [pdf] replaced on 2016-12-20 03:29:10

Proof of the Riemann Hypothesis

Authors: Armando M. Evangelista Jr.
Comments: 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of the zeta funtion ζ(s) are real in the critical strip, 0 ≤ σ ≤ 1; or equivalently, if ζ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.
Category: Number Theory

[602] viXra:1612.0296 [pdf] replaced on 2016-12-19 05:18:45

Proof of the Riemann Hypothesis

Authors: Armando M. Evangelista Jr.
Comments: 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of ξ(s) are real in the critical strip 0 ≤ σ ≤ 1, or equivalently, if ξ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.
Category: Number Theory

[601] viXra:1612.0223 [pdf] replaced on 2016-12-15 14:31:17

The Suggestion that 2-Probable Primes Satisfying Even Goldbach Conjecture Are Possible

Authors: Prashanth R. Rao
Comments: 1 Page.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.
Category: Number Theory

[600] viXra:1612.0042 [pdf] replaced on 2016-12-19 03:14:49

Proof of Twin Prime Conjecture

Authors: Safa Abdallah Moallim
Comments: 8 Pages.

In this paper we prove that there exist infinitely many twin prime numbers by studying n when 6n ± 1 are primes. By studying n we show that for every n that generates a twin prime number, there has to be m > n that generates a twin prime number too.
Category: Number Theory

[599] viXra:1611.0390 [pdf] replaced on 2016-12-08 03:13:44

Proof of Bunyakovsky's Conjecture

Authors: Robert Deloin
Comments: 10 Pages. This is version 2 with important changes.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate innitely many primes. The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences. As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and the binary Goldbach conjecture. The method is also used to prove that there are infinitely many primorial and factorial primes.
Category: Number Theory

[598] viXra:1610.0065 [pdf] replaced on 2016-10-10 23:28:04

Lauricella Hypergeometric Series Over Finite Fields

Authors: Bing He
Comments: 22 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. Some of these generalize some known results of Li \emph{et al} as well as several other well-known results.
Category: Number Theory