Number Theory

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2007 - 0703(3) - 0706(2)
2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)
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2012 - 1201(2) - 1202(13) - 1203(7) - 1204(9) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(15) - 1211(10) - 1212(4)
2013 - 1301(5) - 1302(10) - 1303(16) - 1304(15) - 1305(12) - 1306(13) - 1307(26) - 1308(12) - 1309(9) - 1310(13) - 1311(16) - 1312(21)
2014 - 1401(20) - 1402(11) - 1403(23) - 1404(11) - 1405(17) - 1406(21) - 1407(35) - 1408(52) - 1409(47) - 1410(18) - 1411(16)

Recent submissions

Any replacements are listed further down

[834] 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

[833] 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

[832] 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

[831] 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

[830] 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

[829] 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

[828] 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

[827] 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

[826] 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

[825] 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

[824] 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

[823] 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

[822] viXra:1411.0048 [pdf] submitted on 2014-11-07 01:55:50

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

[821] viXra:1411.0020 [pdf] submitted on 2014-11-04 02:29:34

Few Notable Observations on the Prime Factors of the Fibonacci Numbers Involving Deconcatenation and Congruence Modulo

Authors: Marius Coman
Comments: 4 Pages.

In one of my previous papers, namely “A conjecture about a large subset of Carmichael numbers related to concatenation”, I obtained interesting results combining the method of deconcatenation with the method of congruence modulo. Applying the same methods to the prime factors of the Fibonacci numbers I found also notable patterns.
Category: Number Theory

[820] viXra:1411.0011 [pdf] submitted on 2014-11-02 08:12:19

Infinitely Prime Proof

Authors: T.Nakashima
Comments: 1 Page.

A doubt of the classialy well known proof.
Category: Number Theory

[819] viXra:1410.0174 [pdf] submitted on 2014-10-27 11:29:15

Solving Diophantine Equations

Authors: Octavian Cira, Florentin Smarandache
Comments: 252 Pages.

In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation η(π(x)) = π(η(x)), where η is the Smarandache function and π is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful ”tools” for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.
Category: Number Theory

[818] viXra:1410.0142 [pdf] submitted on 2014-10-23 05:19:00

Invariant Relations Between Binary Goldbach’s Decompositions’numbers Coded in a 4 Letters Language

Authors: Denise Vella-Chemla
Comments: 23 Pages.

We propose a modelization of binary Goldbach's decompositions in a 4 letters language that permits to envisage this problem in a new way.
Category: Number Theory

[817] viXra:1410.0140 [pdf] submitted on 2014-10-23 05:48:34

Two Exciting Classes of Odd Composites Defined by a Relation Between Their Prime Factors

Authors: Marius Coman
Comments: 3 Pages.

In this paper I will define two interesting classes of odd composites often met (by the author of this paper) in the study of Fermat pseudoprimes, which might also have applications in the study of big semiprimes or in other fields. This two classes of composites n = p(1)*...*p(k), where p(1), ..., p(k) are the prime factors of n are defined in the following way: p(j) – p(i) + 1 is a prime or a power of a prime, respectively p(i) + p(j) – 1 is a prime or a power of prime for any p(i), p(j) prime factors of n such that p(1) ≤ p(i) < p(j) ≤ p(k).
Category: Number Theory

[816] viXra:1410.0112 [pdf] submitted on 2014-10-19 23:08:56

An Approach to Explore the Infinite Nature of Twin Primes

Authors: Prashanth R. Rao
Comments: 2 Pages.

Abstract: Twin prime conjecture states that there are infinite number of twin primes of the form p and p+2. Remarkable progress has recently been achieved by Y. Zhang to show that infinite primes that differ by large gap (~ 70 million) exist and this gap has been further narrowed to ~600 by others. We use an elementary approach to explore any obvious constraint that could limit the infinite nature of twin primes. Using Fermat’s little theorem as a surrogate for primality we derive an equation that suggests but not prove that twin primes can be infinite.
Category: Number Theory

[815] viXra:1410.0108 [pdf] submitted on 2014-10-19 11:45:01

A Simple Criterion for Finding Mersenne Primes

Authors: Zeraoulia Elhadj
Comments: 2 Pages.

In this note, we introduce a simple criterion to prove that a given Mersenne number is really a Mersenne prime.
Category: Number Theory

[814] viXra:1410.0107 [pdf] submitted on 2014-10-19 06:16:32

A Proof of the ABC Conjecture

Authors: Zhang Tianshu
Comments: 16 Pages.

We first get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. Next expound relations between C and paf (ABC) by the symmetric law of odd numbers. Finally we have proven C ≤ Cε [paf (ABC)] 1+ ε such being the case A+B=C, and gcf (A, B, C) =1.
Category: Number Theory

Replacements of recent Submissions

[377] 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

[376] viXra:1411.0206 [pdf] replaced on 2014-11-24 00:22:53

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

Authors: Vincenzo Oliva
Comments: 6 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

[375] 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

[374] viXra:1411.0020 [pdf] replaced on 2014-11-07 15:24:10

Few Notable Observations on the Prime Factors of the Fibonacci Numbers Involving Deconcatenation and Congruence Modulo

Authors: Marius Coman
Comments: 5 Pages.

In one of my previous papers, namely “A conjecture about a large subset of Carmichael numbers related to concatenation”, I obtained interesting results combining the method of deconcatenation with the method of congruence modulo. Applying the same methods to the prime factors of the Fibonacci numbers I found also notable patterns.
Category: Number Theory

[373] viXra:1411.0011 [pdf] replaced on 2014-11-07 01:30:46

Infinitely Prime Proof

Authors: T.Nakashima
Comments: 2 Pages.

A doubt of the classialy well known proof.
Category: Number Theory