Number Theory

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Recent submissions

Any replacements are listed further down

[848] viXra:1412.0178 [pdf] submitted on 2014-12-15 05:30:02

Trigonometrical Form of Zeta-Function for Negative (N) with Proof for Riemann Hypothesis.

Authors: Grzegorz Ileczko
Comments: 13 Pages.

The Riemann hypothesis is not proved by more, than 150 years. At this paper, I presented new solution for this problem. I found new trigonometrical form of Riemann's zeta function for negative numbers (n). This new form of zeta gives opportunity to prove the Riemann hypothesis. Presented proof isn’t complicated for trigonometrical form of zeta function.
Category: Number Theory

[847] viXra:1412.0164 [pdf] submitted on 2014-12-11 15:54:12

Rebuttal of Paper by Matthias Lesch, 13 September 2014, Vixra

Authors: Stephen Marshall
Comments: 5 Pages.

This paper presents a complete rebuttal of the paper Vixra 1408.0195v2 posted by Matthias Lesch on 13 September 2014. This rebuttal is in response to Vixra 1408.0195v2 where Matthias Lesch erroneously attempted to disprove six papers I published proving several conjectures in Number Theory. Specifically, these were papers Vixra:1408.0169, 1408.0174, 1408.0201, 1408.0209, and 1408.0212. This rebuttal paper is presented in the same format as Vixra 1408.0195v2 with necessary quotes from paper Vixra 1408.0195v2 to clarify rebuttals.
Category: Number Theory

[846] viXra:1412.0150 [pdf] submitted on 2014-12-10 02:40:58

The Mar Reduced Form of a Natural Number

Authors: Marius Coman
Comments: 60 Pages. Published in Romania in 1998. Copyright 1998-2003 by publishing house "B.I.C. ALL". Copyright since 2003 by Marius Coman

In this paper I define a function which allows the reduction to any non-null positive integer to one of the digits 1, 2, 3, 4, 5, 6, 7, 8 or 9. The utility of this enterprise is well-known in arithmetic; the function defined here differs apparently insignificant but perhaps essentially from the function modulo 9 in that is not defined on 0, also can’t have the value 0; essentially, the mar reduced form of a non-null positive integer is the digital root of this number expressed as a function such it can be easily used in various applications (divizibility problems, diophantine equations), defined only on the operations of addition and multiplication not on the operations of subtraction and division. One of the results obtained with this tool is, as I know, the first proof of Fermat’s last Theorem, case n = 3, using just integers, no complex numbers (it is known that Fermat proved himself the case n = 4 and many proofs for this case there exist using only integers but I do not know one for case n = 3).
Category: Number Theory

[845] viXra:1412.0136 [pdf] submitted on 2014-12-07 03:30:07

A Proof of Strong Goldbach Conjecture and Twin Prime Conjecture

Authors: Pingyuan Zhou
Comments: 16 Pages. This paper has been submitted to mathematical journal.

In this paper we give a proof of the strong Goldbach conjecture by studying limit status of original continuous Goldbach natural number sequence generated by original continuous odd prime number sequence. It implies the weak Goldbach conjecture. If a prime p is defined as Goldbach prime when GNL = p then Goldbach prime is the higher member of a twin prime pair, from which we will give a proof of the twin prime conjecture.
Category: Number Theory

[844] viXra:1412.0124 [pdf] submitted on 2014-12-06 03:01:39

Sieve of Prime Numbers Using Algorithm

Authors: Barar Stelian Liviu
Comments: 20 Pages.

If in the Sieve of Eratosthenes the majority of multiplication of prime numbers result in a results devoid of practical benefit (numbers divisible by 2,3 and/or 5) , in the sieve of prime numbers , each multiplication of prime number gives a result in a number not divisible to 2,3 and/or 5.
Category: Number Theory

[843] viXra:1412.0111 [pdf] submitted on 2014-12-05 03:52:06

The Twin Prime Theorem

Authors: Jian Ye
Comments: 4 Pages.

The paper from the prime origin,derived the equations of new prime number theorem and the twin prime theorem,and it revealed the equivalence between the generalized twin primes.
Category: Number Theory

[842] viXra:1412.0046 [pdf] submitted on 2014-12-03 04:34:06

Two Conjectures on Sequences of Primes Obtained from the Lesser Term from a Pair of Twin Primes

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make two conjectures about two types of possible infinite sequences of primes obtained starting from any given prime which is the lesser term from a pair of twin primes for a possible infinite of positive integers which are not of the form 3*k – 1 respectively starting from any given positive integer which is not of the form 3*k - 1 for a possible infinite of lesser terms from pairs of twin primes.
Category: Number Theory

[841] viXra:1412.0044 [pdf] submitted on 2014-12-02 22:18:18

A Proof of the Beal’s Conjecture (After Fourth Modification)

Authors: Zhang Tianshu
Comments: 26 Pages.

First, we classify A, B and C according to their respective odevity, and ret rid of two kinds from AX+BY=CZ. Then, affirm AX+BY=CZ such being the case A, B and C have a common prime factor by concrete examples. After that, prove AX+BY≠CZ such being the case A, B and C have not any common prime factor by the mathematical induction with the aid of the symmetric law of odd numbers after the decomposition of the inequality. Finally, reached such a conclusion that the Beal’s conjecture can hold water after the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.
Category: Number Theory

[840] viXra:1412.0042 [pdf] submitted on 2014-12-03 02:29:23

Eight Conjectures on Chameleonic Numbers Involving a Formula Based on the Multiples of 30

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make eight conjectures about a type of numbers which I defined in a previous paper, “The notion of chameleonic numbers, a set of composites that «hide» in their inner structure an easy way to obtain primes”, in the following way: the non-null positive composite squarefree integer C not divisible by 2, 3 or 5 is such a number if the absolute value of the number P – d + 1 is always a prime or a power of a prime, where d is one of the prime factors of C and P is the product of all prime factors of C but d.
Category: Number Theory

[839] viXra:1412.0041 [pdf] submitted on 2014-12-03 03:26:52

Lucasian Primality Criterion for Specific Class of K*2^n-1

Authors: Predrag Terzic
Comments: 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k*2^n-1 is introduced .
Category: Number Theory

[838] viXra:1412.0039 [pdf] submitted on 2014-12-02 12:33:58

Eight Conjectures on a Certain Type of Semiprimes Involving a Formula Based on the Multiples of 30

Authors: Marius Coman
Comments: 4 Pages.

In this paper I make four conjectures about a certain type of semiprimes which I defined in a previous paper, “Two exciting classes of odd composites defined by a relation between their prime factors”, in the following way: Coman semiprimes of the first kind are the semiprimes p*q with the property that q1 – p1 + 1 = p2*q2, where the semiprime p2*q2 has also the property that q2 – p2 + 1 = p3*q3, also a semiprime, and the operation is iterate until eventually pk – qk + 1 is a prime. I also defined Coman semiprimes of the second kind the semiprimes p*q with the property that q1 + p1 - 1 = p2*q2, where the semiprime p2*q2 has also the property that q2 + p2 - 1 = p3*q3, also a semiprime, and the operation is iterate until eventually pk + qk - 1 is a prime.
Category: Number Theory

[837] viXra:1412.0036 [pdf] submitted on 2014-12-02 10:12:25

Two Formulas of Generalized Fermat Numbers Which Seems to Generate Large Primes

Authors: Marius Coman
Comments: 2 Pages.

There exist few distinct generalizations of Fermat numbers, like for instance numbers of the form F(k) = a^(2^k) + 1, where a > 2, or F(k) = a^(2^k) + b^(2^k) or Smarandache generalized Fermat numbers, which are the numbers of the form F(k) = a^(b^k) + c, where a, b are integers greater than or equal to 2 and c is integer such that (a, c) = 1. In this paper I observe two formulas based on a new type of generalized Fermat numbers, which are the numbers of the form F(k) = (a^(b^k) ± c)/d, where a, b are integers greater than or equal to 2 and c, d are positive non-null integers such that F(k) is integer.
Category: Number Theory

[836] viXra:1411.0579 [pdf] submitted on 2014-11-27 09:52:30

A Useful Criterion to Identify Candidate Twin Primes

Authors: Prashanth R. Rao
Comments: 2 Pages.

In a previous paper we derived that if p, p+2 are twin-primes then 2p-2 is of the form (pz+y) where z, y must have unique solutions. We extend this result to derive a single criterion that we believe is novel that may be useful to screen for candidate twin primes.
Category: Number Theory

[835] viXra:1411.0571 [pdf] submitted on 2014-11-27 03:35:47

Recurrent Formulas Which Conduct to Probably Infinite Sequences of Primes and a Generalization of a Cunningham Chain

Authors: Marius Coman
Comments: 4 Pages.

In this paper I define few formulas which conduct from any odd prime respectively from any pair of distinct odd primes to an infinity of probably infinite sequences of primes, also to such sequences of a certain kind of semiprimes, and I also make a generalization of a Cunningham chain of primes of the first kind, respectively of the second kind.
Category: Number Theory

[834] viXra:1411.0569 [pdf] submitted on 2014-11-26 09:36:47

A Probably Infinite Sequence of Primes Formed Using Carmichael Numbers, the Number 584 and Concatenation

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make a conjecture which states that there exist an infinity of primes of the form N/3^m, where m is positive integer and N is the number formed concatenating to the left a Carmichael number with the number 584. Such primes are 649081 = 5841729/3^2, 1947607 = 5842821/3, 1948867 = 5846601/3 etc. I also make few comments about a certain kind of semiprimes.
Category: Number Theory

[833] viXra:1411.0563 [pdf] submitted on 2014-11-26 05:00:31

Proof of the Riemann Hypothesis and an Upper Bound for the Divisor Sum Function

Authors: Vincenzo Oliva
Comments: 7 Pages.

Robin's theorem asserts that Robin's inequality (RI) sum_{d|n}d := sigma(n) < e^{gamma} n loglog n, where gamma is the Euler-Mascheroni constant and n>5040, is equivalent to the Riemann Hypothesis. We prove by contradiction there are no counterexamples to RI, consequently providing an immediately ensuing sharper upper bound for sigma(n), odd n.
Category: Number Theory

[832] viXra:1411.0545 [pdf] submitted on 2014-11-22 21:46:43

Conjecture Which States that an Iterative Operation of Any Pair of Two Odd Primes Conducts to a Larger Prime

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make a conjecture which states that from any two odd primes p1 and p2 can be obtained, through an iterative and very simple operation, a prime p3 larger than p1 and also larger than p2.
Category: Number Theory

[831] viXra:1411.0539 [pdf] submitted on 2014-11-22 08:38:35

All Pythagorean Triple Formula

Authors: Maik Becker-Sievert
Comments: 1 Page.

Formula to generate all Pythagorean Triple
Category: Number Theory

[830] viXra:1411.0537 [pdf] submitted on 2014-11-22 04:45:28

All Pythagorean Triple Formulas

Authors: Maik Becker-Sievert
Comments: 1 Page.

Pythagorean Triple Formulas
Category: Number Theory

[829] viXra:1411.0481 [pdf] submitted on 2014-11-20 14:02:24

An Interesting Class of Smarandache Generalized Fermat Numbers

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make few observations on a class of Smarandache generalized Fermat numbers, which are the numbers of the form F(k) = a^(b^k) + c, where a, b are integers greater than or equal to 2 and c is integer such that (a, c) = 1. The class that is observed in this paper includes the numbers of the form F(k) = m^(n^k) + n, where k is positive integer and m and n are coprime positive integers, not both of them odd or both of them even.
Category: Number Theory

[828] viXra:1411.0436 [pdf] submitted on 2014-11-20 06:15:53

Conjectures on Smarandache Generalized Fermat Numbers

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make few conjectures on few classes of generalized Fermat numbers, i.e. the numbers of the form F(k) = 2^(2^k) + n, where k is positive integer and n is an odd number, the numbers of the form F(k) = 4^(4^k) + 3 and the numbers of the form F(k) = m^(m^k) + n, where m + n = p, where p is prime, all subclasses of Smarandache generalized Fermat numbers, i.e. the numbers of the form F(k) = a^(b^k) + c, where a, b are integers greater than or equal to 2 and c is integer such that (a, c) = 1.
Category: Number Theory

[827] viXra:1411.0370 [pdf] submitted on 2014-11-19 11:03:33

Prime Powers of (p − 1) Modulo P^k

Authors: William Wu ChengYuan
Comments: 2 Pages.

We prove that (p-1)^(p^k) is congruent to -1 modulo p^k, if p is a prime, using the Binomial Theorem and Legendre's Theorem.
Category: Number Theory

[826] viXra:1411.0109 [pdf] submitted on 2014-11-13 15:02:53

Why there Are no Odd Perfect Numbers

Authors: A. Garces Doz
Comments: 9 Pages.

In this study we propose a demonstration of the impossibility of odd perfect numbers.This proof uses a congruence, which is implicit in the condition, mandatory, demonstrated by Euler. More precisely, a congruence that must be fulfilled in the equation that equals the number 2N, with Euler condition, and the formula for the sum of the divisors of the odd number N. Following a rigorous and meticulous way, this mandatory congruence; a final equation is obtained after one polynomial simplification on both sides of the original equation that equals the number 2N with the sum of the divisors of the number N. With this final equation, the impossibility of the existence of odd perfect numbers is demonstrated by applying several lemmas.These lemmas are demonstrations already established by W. Ljunggren, Maohua Le, Nagell, among others. With a lemma that establishes mandatory requirements, and two other lemmas for the absence of solutions on certain specific Diophantine equations ( (x^n - 1)/(x-1) = y^2 ; (x^n + 1)/(x+1) = y^2 ; n = 2z + 1); ultimately lead to the demonstration of the nonexistence of odd perfect numbers.
Category: Number Theory

[825] viXra:1411.0084 [pdf] submitted on 2014-11-10 11:32:55

The Cicada of a + B = C

Authors: Th.G.
Comments: 1 Page.

Why the ABC-Conjecture never holds by tom.gu8@gmail.com
Category: Number Theory

[824] viXra:1411.0075 [pdf] submitted on 2014-11-08 23:05:43

Riemann Hypothesis: An Original Approach

Authors: A. A. Frempong
Comments: 9 Pages. Copyright © A. A. Frempong

Assuming the sum of the original Riemann series is L, a ratio method was used to split-up the series equation into sub-equations and each sub-equation was solved in terms of L, and ratio terms. It is to be noted that unquestionably, each term of the series equation contributes to the sum, L, of the series. There are infinitely many sub-equations and solutions corresponding to the infinitely many terms of the series equation. After the sum, L, and the ratio terms have been determined and substituted in the corresponding equations, the Riemann hypothesis would surely be either proved or disproved, since the original equation is being solved. Using the original series equation eliminates possible hidden flaws in derived equations and consequent solutions.
Category: Number Theory

[823] viXra:1411.0071 [pdf] submitted on 2014-11-08 14:46:15

Formula that Uses Primes as Input Values for Obtaining Larger Primes, Based on the Numbers 7 and 186

Authors: Marius Coman
Comments: 2 Pages.

In this paper I present a formula, based on the numbers 7 and 186, that, using primes as input values, often leads, as output values, to larger primes, also to squares of primes and semiprimes. I found this formula by chance, playing with two of my favourite numbers, 13 and 31, and observing that 7*13^2 + 6*31 = 37^2 (to be noted, without necessarily connection with this paper, that the difference between the two known Wieferich primes, 1093 and 3511, is equal to 6*13*31).
Category: Number Theory

[822] viXra:1411.0069 [pdf] submitted on 2014-11-08 11:54:38

A Formula for Generating a Certain Kind of Semiprimes Based on the Two Known Wieferich Primes

Authors: Marius Coman
Comments: 3 Pages.

In one of my previous papers, “A possible infinite subset of Poulet numbers generated by a formula based on Wieferich primes” I pointed an interesting relation between Poulet numbers and the two known Wieferich primes (not the known fact that the squares of these two primes are Poulet numbers themselves but a way to relate an entire set of Poulet numbers by a Wieferich prime). Exploring further that formula I found a way to generate primes, respectively semiprimes of the form q1*q2, where q2 – q1 is equal to a multiple of 30.
Category: Number Theory

[821] viXra:1411.0065 [pdf] submitted on 2014-11-07 17:04:01

Formulas that Generate Large Primes and Products of Very Few Primes Based on Carmichael Numbers and the Number 375

Authors: Marius Coman
Comments: 2 Pages.

In this paper I combine my interest for Carmichael numbers with my interest for finding formulas that generate large primes or products of very few primes showing few easy ways for obtaining such numbers and at the same time an interesting relation between absolute Fermat pseudoprimes and the number 375.
Category: Number Theory

Replacements of recent Submissions

[384] viXra:1412.0041 [pdf] replaced on 2014-12-03 08:13:58

Lucasian Primality Criterion for Specific Class of K*2^n-1

Authors: Predrag Terzic
Comments: 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k*2^n-1 is introduced .
Category: Number Theory

[383] viXra:1411.0563 [pdf] replaced on 2014-12-01 12:05:53

Proof of the Riemann Hypothesis and an Upper Bound for the Divisor Sum Function

Authors: Vincenzo Oliva
Comments: 8 Pages.

Robin's theorem asserts that Robin's inequality (RI) sum_{d|n}d := sigma(n) < e^{gamma} n loglog n, where gamma is the Euler-Mascheroni constant and n>5040, is equivalent to the Riemann Hypothesis. We prove by contradiction there are no counterexamples to RI, consequently providing an immediately ensuing sharper upper bound for sigma(n), odd n.
Category: Number Theory

[382] viXra:1411.0436 [pdf] replaced on 2014-11-20 13:26:16

Conjectures on Smarandache Generalized Fermat Numbers

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make few conjectures on few classes of generalized Fermat numbers, i.e. the numbers of the form F(k) = 2^(2^k) + n, where k is positive integer and n is an odd number, the numbers of the form F(k) = 4^(4^k) + 3 and the numbers of the form F(k) = m^(m^k) + n, where m + n = p, where p is prime, all subclasses of Smarandache generalized Fermat numbers, i.e. the numbers of the form F(k) = a^(b^k) + c, where a, b are integers greater than or equal to 2 and c is integer such that (a, c) = 1.
Category: Number Theory

[381] viXra:1411.0109 [pdf] replaced on 2014-11-14 07:38:24

Why there Are no Odd Perfect Numbers

Authors: A. Garcés Doz
Comments: 9 Pages. Corrected a mistake that does not affect at all the proof

In this study we propose a demonstration of the impossibility of odd perfect numbers.This proof uses a congruence, which is implicit in the condition, mandatory, demonstrated by Euler. More precisely, a congruence that must be fulfilled in the equation that equals the number 2N, with Euler condition, and the formula for the sum of the divisors of the odd number N. Following a rigorous and meticulous way, this mandatory congruence; a final equation is obtained after one polynomial simplification on both sides of the original equation that equals the number 2N with the sum of the divisors of the number N. With this final equation, the impossibility of the existence of odd perfect numbers is demonstrated by applying several lemmas.These lemmas are demonstrations already established by W. Ljunggren, Maohua Le, Nagell, among others. With a lemma that establishes mandatory requirements, and two other lemmas for the absence of solutions on certain specific Diophantine equations ( (x^n - 1)/(x-1) = y^2 ; (x^n + 1)/(x+1) = y^2 ; n = 2z + 1); ultimately lead to the demonstration of the nonexistence of odd perfect numbers.
Category: Number Theory