**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(8) - 1203(6) - 1204(8) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(51) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(15) - 1508(14) - 1509(14) - 1510(12) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(19) - 1608(16) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(24) - 1703(28) - 1704(32) - 1705(25) - 1706(26) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(24)

Any replacements are listed further down

[1624] **viXra:1711.0353 [pdf]**
*submitted on 2017-11-19 03:41:41*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: Any square of a prime p^2, where p > 3, can be written as p + q + (n*q – n + 1) or as p + q + (n*q - n – 1), where q and n*q – n + 1 respectively n*q - n – 1 are primes and n positive integer. Examples: 11^2 = 121 = 11 + 37 + (2*37 – 1), where 37 and 2*37 – 1 = 73 are primes; 13^2 = 169 = 13 + 53 + (2*53 – 3), where 53 and 2*53 – 3 = 103 are primes. An equivalent formulation of the conjecture is that for any prime p, p > 3, there exist n positive integer such that one of the numbers q = (p^2 – p + n – 1)/(n + 1) or q = p^2 – p + n + 1)/(n + 1) is prime satisfying also the condition that p^2 – p – q is prime.

**Category:** Number Theory

[1623] **viXra:1711.0343 [pdf]**
*submitted on 2017-11-18 03:29:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

Playing with Carmichael numbers, a set of numbers I’ve always been fond of (I’ve “discovered” Fermat’s “Little” Theorem and the first few Carmichael numbers before I know they had already been discovered), I noticed that the formula C + 81*2^(4*d), where C is a Carmichael number and d one of its prime factors, gives often primes or products of very few primes. For instance, for C = 1493812621027441 are obtained in this manner three primes: 2918779690625137, 6729216728661136606577017055290271857 and 644530914387083488233375393598279808770191171433362641802841314053534708129737067311868017 (a 90-digit prime!), respectively for d = 11, d = 29 and d = 73.

**Category:** Number Theory

[1622] **viXra:1711.0330 [pdf]**
*submitted on 2017-11-17 01:34:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1 and dn be the least respectively the largest prime factors of the number P1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form P1 + |P1 – dn^2|*d1, where |P1 – dn^2| is the absolute value of P1 – dn^2. Example: for Poulet number P1 = 1729 = 7*13*19 is obtained through this operation Poulet number P2 = 11305 (1729 – 19^2 = 1368 and 1729 + 1368*7 = 11305). Note that from 11 from the first 30 Poulet numbers (P1) were obtained through this method Poulet numbers (P2).

**Category:** Number Theory

[1621] **viXra:1711.0307 [pdf]**
*submitted on 2017-11-14 06:41:14*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas involving pi and G (Catalan constant).

**Category:** Number Theory

[1620] **viXra:1711.0303 [pdf]**
*submitted on 2017-11-14 06:51:50*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some elementary integrals for pi.

**Category:** Number Theory

[1619] **viXra:1711.0296 [pdf]**
*submitted on 2017-11-14 04:44:03*

**Authors:** Kurmet Sultan

**Comments:** 9 Pages.

Earlier in [http://vixra.org/abs/1708.0177] the author presented a proof of the Collatz conjecture, based on the regularities of numbers of the form 6n ± 1, formed as a result of calculating the Collatz function. In [http://vixra.org/abs/1708.0177] there were many tables, figures, definitions, examples and explanations, which created difficulties in the perception of the material. Taking this into account, in this paper we give a shortened version of the proof of the Collatz conjecture.

**Category:** Number Theory

[1618] **viXra:1711.0291 [pdf]**
*submitted on 2017-11-12 10:24:36*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages.

This article simplifies Niven and others proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores unified approach to classic irrationality and transcendence proofs.

**Category:** Number Theory

[1617] **viXra:1711.0283 [pdf]**
*submitted on 2017-11-12 23:30:43*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved.

A sentence to test is if known zeroes imply other zeroes. This effectively tests if a location of zeroes (trivial based on even numbers) and a location of zeroes (non trivial based on odd numbers) imply possibly another location of zeroes as a tautology, because the question is "Are there possibly other zeroes".

**Category:** Number Theory

[1616] **viXra:1711.0276 [pdf]**
*submitted on 2017-11-11 13:07:08*

**Authors:** Dariusz Dudało

**Comments:** 1 Page.

Monty Hall problem

**Category:** Number Theory

[1615] **viXra:1711.0267 [pdf]**
*submitted on 2017-11-10 23:39:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: The square of any odd prime can be obtained from the numbers of the form 360*k + 72 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number 360*k + 72; than for any square of a prime p^2 there exist k such that (d1 - 1)*(d2 - 1)*...*(dn - 1) + 1 = p^2. Example: for p^2 = 13^2 = 169 there exist k = 17 such that from 360*17 + 72 = 6192 = 2^4*3^2*43 is obtained 1^4*2^2*42 + 1 = 169. I also conjecture that any absolute Fermat pseudoprime (Carmichael number) can be obtained through the presented formula, which attests again the special relation that I have often highlighted between the nature of Carmichael numbers and the nature of squares of primes.

**Category:** Number Theory

[1614] **viXra:1711.0262 [pdf]**
*submitted on 2017-11-10 11:00:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number P1 – 1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form (d1 + 1)*(d2 + 1)*...*(dn + 1) + 1. Example: for Poulet number P1 = 645 is obtained through this operation Poulet number P2 = 1729 (644 = 2*2*7*23 and 3*3*8*24 + 1 = 1729). Note that from more than one Poulet number P1 can be obtained the same Poulet number P2 (from both 1729 and 6601 is obtained 46657).

**Category:** Number Theory

[1613] **viXra:1711.0258 [pdf]**
*submitted on 2017-11-09 13:06:12*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.

**Category:** Number Theory

[1612] **viXra:1711.0249 [pdf]**
*submitted on 2017-11-08 09:49:05*

**Authors:** I. N. Tukaev

**Comments:** 3 Pages.

This paper proves that the Dirichlet series determining the Riemann zeta function converges within a domain of a real component of complex variable equal to one, with an imaginary component non-equal to zero.

**Category:** Number Theory

[1611] **viXra:1711.0247 [pdf]**
*submitted on 2017-11-07 09:41:06*

**Authors:** Edigles Guedes

**Comments:** 14 Pages.

We demonstrate some elementary identities for quocient of q-series.

**Category:** Number Theory

[1610] **viXra:1711.0239 [pdf]**
*submitted on 2017-11-07 03:53:46*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents two BBP-type formulas

**Category:** Number Theory

[1609] **viXra:1711.0236 [pdf]**
*submitted on 2017-11-06 18:00:00*

**Authors:** Edigles Guedes

**Comments:** 16 Pages.

We demonstrate some elementary identities for q-series involving the q-Pochhammer symbol, as well as an identity involving the generating functions of the (m,k)-capsids and (m, r1, r2)-capsids.

**Category:** Number Theory

[1608] **viXra:1711.0203 [pdf]**
*submitted on 2017-11-05 20:45:06*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds which belong not to AX+BY=CZ. Then, affirm AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, prove AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric law of odd numbers whereby even number 2W-1HZ as symmetric center after divide the inequality in four. Finally, reach a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[1607] **viXra:1711.0202 [pdf]**
*submitted on 2017-11-06 02:56:59*

**Authors:** Kunle Adegoke

**Comments:** 17 Pages.

We present new infinite arctangent sums and infinite sums of products of arctangents. Many previously known evaluations appear as special cases of the general results derived in this paper.

**Category:** Number Theory

[1606] **viXra:1711.0140 [pdf]**
*submitted on 2017-11-04 16:02:17*

**Authors:** José de Jesús Camacho Medina

**Comments:** 3 Pages.

This article disseminates a series of new and interesting mathematical formulas for the fibonacci sequence as product of the investigations of the author since 2015.

**Category:** Number Theory

[1605] **viXra:1711.0134 [pdf]**
*submitted on 2017-11-05 04:46:28*

**Authors:** Philip Gibbs, Judson McCranie

**Comments:** 8 Pages.

All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps.

**Category:** Number Theory

[1604] **viXra:1711.0130 [pdf]**
*submitted on 2017-11-03 10:35:22*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. Suitable for first year, first term calculus students: just derivatives of polynomials.

Using just the derivative of the sum is the sum of the derivatives a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.

**Category:** Number Theory

[1603] **viXra:1711.0128 [pdf]**
*submitted on 2017-11-03 22:24:48*

**Authors:** Choe Ryujin

**Comments:** 4 Pages.

Theorem of prime pairs

**Category:** Number Theory

[1602] **viXra:1711.0127 [pdf]**
*submitted on 2017-11-03 23:29:58*

**Authors:** Bado idriss olivier

**Comments:** 7 Pages.

In this paperwe give the proof Polignac Conjecture
by using Chebotarev -Artin theorem ,Mertens formula and Poincaré sieve For doing that we prove that .Let's X be an arbitrarily large real number and n an even integer we prove that there are many primes p such that p+n is prime between sqrt(X) and X

**Category:** Number Theory

[1601] **viXra:1711.0109 [pdf]**
*submitted on 2017-11-02 11:47:31*

**Authors:** Antoine Balan

**Comments:** 5 Pages.

We present here the qq'-caculus which generalize the q-calculus but is however limited.

**Category:** Number Theory

[1600] **viXra:1710.0353 [pdf]**
*submitted on 2017-10-30 19:15:50*

**Authors:** José de Jesús Camacho Medina

**Comments:** 6 Pages.

This article disseminates a series of new and interesting mathematical formulas, there are formulas of prime numbers, fibonacci sequence, square root and others as product of the investigations of the author since 2011.

**Category:** Number Theory

[1599] **viXra:1710.0348 [pdf]**
*submitted on 2017-10-31 03:21:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures on Novák-Carmichael numbers: (1) There exist an infinity of Novák-Carmichael numbers of the form (30n + p)*(30n + q) – p*q for any [p, q] distinct primes of the form 6k + 1; (2) There exist an infinity of Novák-Carmichael numbers of the form (30n + p)*(30n + q) - p*q for any [p, q] distinct primes of the form 6k – 1, where k > 1. See the sequence A124240 in OEIS for Novák-Carmichael numbers (numbers n such that a^n ≡ 1 (mod n) for every a coprime to n).

**Category:** Number Theory

[1598] **viXra:1710.0347 [pdf]**
*submitted on 2017-10-31 03:23:06*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following three conjectures on Novák-Carmichael numbers: (1) There exist an infinity of Novák-Carmichael numbers of the form (6k + 1)*(12k + 1)*(18k + 1) – 1; (2) There exist an infinity of Novák-Carmichael numbers of the form (6k - 1)*(12k - 1)*(18k - 1) + 1; (3) There exist an infinity of Novák-Carmichael numbers C such that C + 1 is a Poulet number. See the sequence A124240 in OEIS for Novák-Carmichael numbers (numbers n such that a^n ≡ 1 (mod n) for every a coprime to n).

**Category:** Number Theory

[1597] **viXra:1710.0339 [pdf]**
*submitted on 2017-10-31 19:02:00*

**Authors:** Andrei Lucian Dragoi

**Comments:** 32 Pages.

This article proposes a synthesized classification of some Goldbach-like conjectures, including those which are "stronger" than the Binary Goldbach's Conjecture (BGC) and launches a new generalization of BGC briefly called "the Vertical Binary Goldbach's Conjecture" (VBGC), which is essentially a meta-conjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on "iterative" primes with recursive prime indexes (i-primeths). VBGC was discovered by the author of this paper in 2007 and perfected (by computational verifications) until 2017 by using the arrays of matrices of Goldbach index-partitions, which are a useful tool in studying BGC by focusing on prime indexes. VBGC distinguishes as a very important conjecture of primes, with potential importance in the optimization of the BGC experimental verification (including other possible theoretical and practical applications in mathematics and physics) and a very special self-similar property of the primes set. Keywords: Primes with prime indexes; i-primeths; the Binary Goldbach Conjecture; Goldbach-like conjectures; the Vertical Binary Goldbach Conjecture. 2010 mathematics subject classification: 11N05 (Distribution of primes, URL: http://www.ams.org/msc/msc2010.html?t=11N05&btn=Current) OFFICIAL LINKS OF THIS PUBLISHED (OPEN) PEER-REVIEWED ARTICLE: http://www.sciencedomain.org/issue/3151 http://www.journalrepository.org/media/journals/JAMCS_69/2017/Oct/Andrei2522017JAMCS36895.pdf http://www.sciencedomain.org/review-history/21625 http://www.sciencedomain.org/metrics/21625

**Category:** Number Theory

[1596] **viXra:1710.0335 [pdf]**
*submitted on 2017-10-29 21:39:34*

**Authors:** Lulu Karami

**Comments:** 17 Pages.

This submission is more or less an amateur exposition on a specific elliptic curve, discussing counting points over finite fields as well as constructing an associated $L$-function and pinning down the affiliated special value $L(E, 1)$ for the elliptic curve $E$ primarily discussed throughout this piece. The techniques and tools presented can be carried over to infinitely many elliptic curves partitioned into two sets depending on 'twists' of two specific curves; one of which happens to be the curve previously, and vaguely, mentioned.

**Category:** Number Theory

[1595] **viXra:1710.0333 [pdf]**
*submitted on 2017-10-30 05:47:22*

**Authors:** José Francisco García Juliá

**Comments:** 2 Pages.

It is obtained a solution of the Fermat’s last theorem.

**Category:** Number Theory

[1594] **viXra:1710.0332 [pdf]**
*submitted on 2017-10-30 07:53:11*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

This note presents three limits for 1/pi

**Category:** Number Theory

[1593] **viXra:1710.0331 [pdf]**
*submitted on 2017-10-30 07:57:53*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a simple formula for pi

**Category:** Number Theory

[1592] **viXra:1710.0263 [pdf]**
*submitted on 2017-10-23 08:03:39*

**Authors:** Edgar Valdebenito

**Comments:** 9 Pages.

This note presents three formulas involving pi and some fractals.

**Category:** Number Theory

[1591] **viXra:1710.0245 [pdf]**
*submitted on 2017-10-22 16:36:56*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 2 Pages. None.

None.

**Category:** Number Theory

[1590] **viXra:1710.0242 [pdf]**
*submitted on 2017-10-22 16:39:39*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 4 Pages. None.

None.

**Category:** Number Theory

[1589] **viXra:1710.0209 [pdf]**
*submitted on 2017-10-18 15:14:30*

**Authors:** Leif R. Uppström, Daniel Uppström

**Comments:** 9 Pages.

In mathematical literature it is asked for a computable function or efficient algorithm to find all, or at least a large subset, of the prime numbers. This paper shows that all primes can be characerised by their reciprocal period length *L* and its figure value *R*. These parameters are given for each prime after inversion to an infinitely repeated period and are used to group all primes into disjoint sets that arise as a function of a geometric progression. This theory suggests new ways to enumerate and find large primes.

**Category:** Number Theory

[1588] **viXra:1710.0205 [pdf]**
*submitted on 2017-10-19 02:50:41*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages.

In this paper it is proved that the sum of consecutive prime numbers under the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, it is proved another asymptotic relationship between the sum of the first prime numbers up to the integer part of the square root of a given natural number and the prime counting function.

**Category:** Number Theory

[1587] **viXra:1710.0174 [pdf]**
*submitted on 2017-10-17 01:35:25*

**Authors:** Choe Ryujin

**Comments:** 4 Pages.

Theorem of prime pair distribution

**Category:** Number Theory

[1586] **viXra:1710.0169 [pdf]**
*submitted on 2017-10-17 09:44:37*

**Authors:** Steven Shawcross

**Comments:** 9 Pages. A version of this paper is copyrighted by Steven Shawcross, 2003.

The integer 2 satisfies the divisibility definition of a prime number: it is only divisible by itself and 1. The integer 1 also satisfies this definition, and yet, mathematicians generally do not consider 1 a prime. Rather 1 merits a class of its own, belonging neither to the prime nor composite class. In divisibility theory, 2 does occupy a special subclass within the class of prime numbers: it is the only even prime. This paper introduces a theory of numbers called the Prime Set Representation Theory. This theory utilizes the odd primes and does not rely on the primeness of 2. In Prime Set Representation Theory, the odd primes are building blocks of the theory; all integers, including 2, have representations in terms of them. The import of the theory is not to dislodge the integer 2 from its solitary, even-prime status. The theory's efficacy is a better understanding of the distribution of primes, twin primes, and primes of the form x^2 + 1. A natural extension of the theory yields valid and strikingly direct approximation formulas for these prime classifications. The same theory furnishes a new and improved approximation to the number of Goldbach pairs associated with general even number 2n (the improvement is relative to Sylvester's formula for Goldbach pairs, but the formula performs well vis-à-vis the Hardy-Littlewood formulas in the ranges tested).

**Category:** Number Theory

[1585] **viXra:1710.0145 [pdf]**
*submitted on 2017-10-12 05:04:05*

**Authors:** Timothy W. Jones

**Comments:** 19 Pages.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(2n+1), n>1, is irrational.

**Category:** Number Theory

[1584] **viXra:1710.0129 [pdf]**
*submitted on 2017-10-11 11:54:50*

**Authors:** Robert C. Hall

**Comments:** 28 Pages.

Regarding Benford's law, many believe that the statistical data sources follow a Benford's law probability density function(1/xLn(10))when, in actuality, it follows a Lognormal probability density function. The only data that strictly follows a Benford's law probability density function is an exponential function i.e. a number (base) raised to a power x. The other sets of data conform to a Lognormal distribution and, as the standard deviation approaches infinity, approximates a true Benford distribution.
Also, the so called Summation theorem whereby the sum of the values with respect to the first digits is a uniform distribution only applies to an exponential function. The data derived from the aforementioned Lognormal distribution is more likely to conform to a Benford like distribution as the data seems to indicate.

**Category:** Number Theory

[1583] **viXra:1710.0113 [pdf]**
*submitted on 2017-10-10 06:32:08*

**Authors:** Kurmet Sultan

**Comments:** 2 Pages. This is the Russian version of the manuscript.

The paper describes the First theorem of Kurmet and a simple proof of the Last theorem of Fermat, which was obtained on the basis of Kurmet's First Theorem.

**Category:** Number Theory

[1582] **viXra:1710.0112 [pdf]**
*submitted on 2017-10-10 06:35:55*

**Authors:** Kurmet Sultan

**Comments:** 2 Pages. This is the Russian version of the manuscript.

In this paper we describe the Second Theorem of Kurmet and give a simple proof of Catalan’s conjecture on the basis of Kurmet's Second Theorem.

**Category:** Number Theory

[1581] **viXra:1710.0109 [pdf]**
*submitted on 2017-10-09 03:05:33*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Fermats Last Theorem n=4
One line proof

**Category:** Number Theory

[1580] **viXra:1710.0099 [pdf]**
*submitted on 2017-10-10 01:50:30*

**Authors:** Zhang Tianshu

**Comments:** 19 Pages.

We first classify all integers ≥2 into eight kinds, and that formulate each of seven kinds therein into a sum of three unit fractions. For remainder one kind, we classify it into three genera, and that formulate each of two genera therein into a sum of three unit fractions. For remainder one genus, we classify it into five sorts, and that formulate each of three sorts therein into a sum of three unit fractions. For remainder two sorts i.e. 4/(49+120c) and 4/(121+120c) with c≥0, we prove them by logical inference. But miss out 3587 concrete fractions to await computer programming to solve the problem that express each of them into a sum of three unit fractions.

**Category:** Number Theory

[1579] **viXra:1710.0048 [pdf]**
*submitted on 2017-10-04 20:55:25*

**Authors:** Choe Ryujin

**Comments:** 1 Page.

Proof of Riemann hypothesis

**Category:** Number Theory

[1578] **viXra:1710.0042 [pdf]**
*submitted on 2017-10-03 11:01:23*

**Authors:** Dieter Sengschmitt

**Comments:** 15 Pages.

I can proof that there are infinitely many twin primes. The twin prime counting function π2(n), which gives the number of twin primes less than or equal to n for any natural number n, is for
limn→∞ π2(n)= 2 C2 [π(n)]^2/n
where π(n) is the prime counting function and C2 is the so-called twin prime constant with C2=0,6601618…

**Category:** Number Theory

[1577] **viXra:1710.0038 [pdf]**
*submitted on 2017-10-03 16:37:46*

**Authors:** Robert C. Hall

**Comments:** 2 Pages.

An attempt is made to derive the probability density function of the sum of prime numbers, which is x/Ln(x). This does appear to be quite accurate in predicting the sum of prime numbers less than 100,000( within 0.124%). Given this assertion, an attempt is made to derive the probability density function of the distribution of the prime numbers themselves.

**Category:** Number Theory

[1576] **viXra:1710.0017 [pdf]**
*submitted on 2017-10-02 02:24:57*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

four new formulas for pi

**Category:** Number Theory

[1575] **viXra:1710.0015 [pdf]**
*submitted on 2017-10-02 03:09:57*

**Authors:** Lulu Karami

**Comments:** 4 Pages.

This submission gives a closed form identity similar to one given by
Ramanujan. A formula for infinitely many similar identities is presented
here as well.

**Category:** Number Theory

[1574] **viXra:1709.0428 [pdf]**
*submitted on 2017-09-28 17:54:57*

**Authors:** Peter Bissonnet

**Comments:** 13 Pages.

This paper again specifies the major points of the article “Do Prime Numbers
Obey a Three-Dimensional Double Helix?” [1] which was received on February
16, 2006 by Hadronic Journal. New information has been added and elucidated
upon, such as why the numbers 2 and 3 are not considered true prime
numbers, and why s in the following formulas for 6s − 1 and for 6s + 1 is really
a composite number equal to the sum of two other numbers, suggesting
that s is always to be considered as an integer. Other new information is added
as well, such as how an engineer in a matter of seconds decomposed a large
prime product into its constituent primes using basic software and won a
contract for his firm.

**Category:** Number Theory

[1573] **viXra:1709.0417 [pdf]**
*submitted on 2017-09-28 08:14:22*

**Authors:** Ramón Ruiz

**Comments:** 26 Pages. This document is written in Spanish

Twin Primes Conjecture: “There are infinitely many primes p such that (p + 2) is also prime”.
In this document I have used the prime numbers theorem enunciated by Carl Friedrich Gauss and the prime numbers theorem in arithmetic progressions. These two theorems applied to a combination of two arithmetic progressions of module 30 that contain prime numbers, allows us to develop a nonprobability general formula to calculate, approximately, the number of prime pairs, p and (p + 2), that are lesser than a number x.
This research is based on a approach designed solely to demonstrate the Twin Prime Conjecture and the Binary Goldbach Conjecture.

**Category:** Number Theory

[1572] **viXra:1709.0411 [pdf]**
*submitted on 2017-09-27 14:14:11*

**Authors:** John Smith

**Comments:** 7 Pages.

Abstract In 1963, a game show called Lets Make A Deal began in the United States. On the show, the host - Monty Hall - would present contestants with the choice of 3 doors, behind only 1 of which was a car. A contestant would pick a door such as No. 1, and Monty, who knew what was behind the doors, would open another door, say No. 2, revealing a goat. Monty would then ask the contestant if they wanted to change their selection to door No. 3. It is widely accepted that the contestant should change doors on the basis that the chances of the car being behind door 3 are 2/3, whereas the chances of the car being behind door 1 are only 1/3. But by appeal to congruities that exist between this seemingly innocuous and simple problem and variety of deeper and less tractable problems, the Monty Hall Problem is revealed as the tip of a great intellectual iceberg.

**Category:** Number Theory

[1571] **viXra:1709.0410 [pdf]**
*submitted on 2017-09-27 14:15:31*

**Authors:** John Smith

**Comments:** 2 Pages.

In 1986 AndrewWiles published a ground-breaking proof of Fermat's Last Theorem. But in spite of the rarity and the significance of the achievement, the underlying reasoning is so convoluted that it would be be extremely difficult -if not impossible- for any but a tiny minority of specialists to understand it. Most must simply take the word of Wiles and his fellow experts that Fermat's Last Theorem has been proved. But the conjecture itself -that no 3 positive integers can satisfy the equation x^n + y^n = z^n for any positive-integer value of n greater than 2- is so simple that a school child could understand it, and Fermat himself claimed that he possessed a proof, one that -if it existed- must have been expressed in the language of 17th century mathematics, and the language of 21st century high school mathematics. Ye there can be no such proof: this note outlines a complimentary but alternative argument to that employed by Wiles that shows why no 17th century proof of the theorem is possible.

**Category:** Number Theory

[1570] **viXra:1709.0408 [pdf]**
*submitted on 2017-09-27 19:45:24*

**Authors:** Johnny E Magee

**Comments:** 18 Pages.

Focusing on the properties and constraints of the decompositions of Fermat’s equation and its elements --and employing only basic arithmetic
and algebraic techniques that would have been known to Fermat-- we identify certain specific requirements necessary for c, of (an + bn) = cn,
to be an integer, and establish that these requirements can only be met at n = 2.
AMS Subject Classification (2010): Primary, 11D41
Key words and phrases: Fermat’s Last Theorem, Fermat’s Equation, Binomial Theorem

**Category:** Number Theory

[1569] **viXra:1709.0375 [pdf]**
*submitted on 2017-09-24 18:23:32*

**Authors:** Matanari Shimoinuda

**Comments:** 28 Pages.

This article is the summary of the spectral interpretation of critical zeroes of an L-function by Alain Connes. I try to examine the subject from the view of the representation theory and add some comments.

**Category:** Number Theory

[1568] **viXra:1709.0312 [pdf]**
*submitted on 2017-09-22 01:36:01*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we analyze the behavior of prime numbers.

**Category:** Number Theory

[1567] **viXra:1709.0295 [pdf]**
*submitted on 2017-09-20 06:46:03*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 4 Pages.

Fermat’s zero theorem is stated as follows: It is impossible to separate a square of a difference of two natural numbers into two squares of differences, or a cube power of a difference into two cube powers of differences, or a fourth power of a difference into two fourth powers, or in general, any power higher than the first, into two like powers of differences.

**Category:** Number Theory

[1566] **viXra:1709.0288 [pdf]**
*submitted on 2017-09-19 04:55:48*

**Authors:** Ranganath G. Kulkarni

**Comments:** 2 Pages.

A quadratic equation for prime numbers is assumed to be true that satisfy the following four rules. Some prime numbers violate these rules. Whereas some non prime numbers satisfy the four rules. They are not prime, therefore to make them violate the fourth rule we need to study how to choose the value of m and n so as make the quadratic equation as the primes generating formula.

**Category:** Number Theory

[1565] **viXra:1709.0258 [pdf]**
*submitted on 2017-09-17 13:33:28*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

**Category:** Number Theory

[1564] **viXra:1709.0257 [pdf]**
*submitted on 2017-09-17 13:35:16*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. French version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

L'égalité de Fermat est contradictoire entre les deuxièmes chiffres des facteurs du nombre A.

**Category:** Number Theory

[1563] **viXra:1709.0256 [pdf]**
*submitted on 2017-09-17 13:36:32*

**Authors:** Victor Sorokine

**Comments:** 2 Pages. Russian version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between
the second digits of the factors of the number А.

Суть противоречия. Равенство Ферма противоречиво по вторым цифрам сомножителей числа А.

**Category:** Number Theory

[1562] **viXra:1709.0227 [pdf]**
*submitted on 2017-09-15 05:23:56*

**Authors:** José Francisco García Juliá

**Comments:** 2 Pages.

It is obtained a minor theorem related with the Fermat conjecture.

**Category:** Number Theory

[1561] **viXra:1709.0128 [pdf]**
*submitted on 2017-09-11 05:10:30*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 3 Pages.

Relationships among natural numbers constituting a Pythagorean triple (PT) and between these natural numbers constituting the Pythagorean triples (PTs) and Prime Numbers (PNs) have been found. These relationships are formulated as theorems; first theorem is that the natural numbers constituting a Pythagorean triple (PT) satisfy a certain equation related to sum of their differences; second theorem is that differences of sum of the natural numbers constituting a Pythagorean triple (PT) are prime numbers.

**Category:** Number Theory

[1560] **viXra:1709.0092 [pdf]**
*submitted on 2017-09-08 12:19:26*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas for pi.

**Category:** Number Theory

[1559] **viXra:1709.0039 [pdf]**
*submitted on 2017-09-04 11:17:48*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Here is defined, following the RSA cryptosystem, a method of cryptography for polynomials over finite rings.

**Category:** Number Theory

[1558] **viXra:1709.0013 [pdf]**
*submitted on 2017-09-02 03:44:43*

**Authors:** Zhang Tianshu

**Comments:** 25 Pages.

Let us regard positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments which have same permutations of c kinds of integers’ points, where c≥1. In this article we shall prove Grimm’s conjecture by the method which changes stepwise symbols of each kind of composite numbers’ points at the original number axis, so as to form consecutive composite numbers’ points inside the limited field of proven Legendre- Zhang conjecture as the true.

**Category:** Number Theory

[1557] **viXra:1709.0003 [pdf]**
*submitted on 2017-09-01 07:17:32*

**Authors:** T.Nakashima

**Comments:** 1 Page.

In this paper, we prove Conway's problem.

**Category:** Number Theory

[1556] **viXra:1708.0421 [pdf]**
*submitted on 2017-08-28 08:10:40*

**Authors:** Edgar Valdebenito

**Comments:** 21 Pages.

This note presents a collection of double integrals for some classical constants.

**Category:** Number Theory

[1555] **viXra:1708.0411 [pdf]**
*submitted on 2017-08-28 10:09:24*

**Authors:** Preininger Helmut

**Comments:** 9 Pages.

In this paper we give an implementation of a Core(c) Number Sieve (for a given c=1,2,3,.. we sift out numbers that have in there factorization a prime with a power >= c). For c=2 we have a squarefree number sieve. (Note, that, for c=1, our implementation compute the usual prime number sieve.) Our goal is to use only one codebase and avoid extra algorithms for every c.
We use some well known algorithms and adopt it for our purpose.

**Category:** Number Theory

[1554] **viXra:1708.0400 [pdf]**
*submitted on 2017-08-28 00:19:33*

**Authors:** Lulu Karami

**Comments:** 48 Pages.

This submission demonstrates how to use the analytic class number formula to express certain quotients of Dedekind's Eta function as a unit raised to the power of a quoteint of class numbers, for particular number fields. It includes a loose derivation for some special cases of reciprocity laws and the Fourier series of particular Eisenstein series.

**Category:** Number Theory

[1553] **viXra:1708.0380 [pdf]**
*submitted on 2017-08-27 09:31:10*

**Authors:** I Gede Putra Yasa ''Gus Satya''

**Comments:** 24 Pages.

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.

**Category:** Number Theory

[1552] **viXra:1708.0255 [pdf]**
*submitted on 2017-08-21 21:17:24*

**Authors:** Jun Chen

**Comments:** 5 Pages.

A new idea of the Goldbach conjecture has been studied, it is that the even number is more bigger, the average form of the sum of two primes are more larger too. And then, we prove that every sufficiently large even number is the sum of two primes.

**Category:** Number Theory

[1551] **viXra:1708.0234 [pdf]**
*submitted on 2017-08-19 11:06:29*

**Authors:** Ramaswamy Krishnan

**Comments:** 3 Pages.

This proof is based on an assumption that value of an infinite series cannot be obtained from a finite number of terms of the series. For all possible factors of (x + y -z) which are not factors of x or y or z, 3 infinite series can be developed, 2 convergent and 1 divergent. In all the 3 cases, the value of the infinite series can be obtained by considering only a finite number of terms. This gives the value for (x + y -z) = p to to the power of alpha * (p1) (p2) (p3). Thus proving Fermat's last theorem.

**Category:** Number Theory

[1550] **viXra:1708.0231 [pdf]**
*submitted on 2017-08-19 11:51:51*

**Authors:** Edigles Guedes, Cícera Guedes

**Comments:** 11 Pages.

In this paper, we construct a relation involving q-Pochhammer symbol, q-bracket, q-factorial and q-binomial coefficient among other things.

**Category:** Number Theory

[1549] **viXra:1708.0230 [pdf]**
*submitted on 2017-08-19 11:56:45*

**Authors:** Edigles Guedes, Cícera Guedes

**Comments:** 4 Pages.

In this paper, we construct a identity for a q-hypergeometric series.

**Category:** Number Theory

[1548] **viXra:1708.0221 [pdf]**
*submitted on 2017-08-19 03:46:20*

**Authors:** Mendzina Essomba Francois

**Comments:** 6 Pages.

I found a formula due to Ramanjan, I have given a generalization in this article

**Category:** Number Theory

[1547] **viXra:1708.0220 [pdf]**
*submitted on 2017-08-19 03:47:39*

**Authors:** Ranganath G. Kulkarni

**Comments:** 1 Page.

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann's formula is approximate one. We need to study the variation of prime numbers with given number x and new variable r.

**Category:** Number Theory

[1546] **viXra:1708.0206 [pdf]**
*submitted on 2017-08-17 05:19:11*

**Authors:** Ramaswamy Krishnan

**Comments:** 3 Pages. Title has been changed a little bit. Instead of mod p, it should be mod p cubed

If\quad { 2 }^{ p-1 }\quad \equiv \quad 1\quad mod\quad ({ p }^{ 3 })\quad then\quad 2,-1,{ 2 }^{ p-2 }\quad are\quad solutions\quad to\quad the\quad equation\\ f(a)\quad =\quad 1\quad -\quad { a }^{ p }\quad -\quad { (1-a) }^{ p\quad \quad }\equiv \quad o\quad mod({ p }^{ 3 }).\quad Using\quad this\quad fact\quad and\quad an\quad expression\quad for\\ { (x+y) }^{ n }\quad \quad in\quad terms\quad of\quad xy\quad ,\quad (x+y)\quad ,\quad ({ x }^{ 2 }+xy+{ y }^{ 2 })\quad it\quad is\quad prooved\quad that\\ { 2 }^{ p-1 }\quad \ncong \quad 1\quad mod({ p }^{ 3 })\quad for\quad any\quad prime\quad 'p'.

**Category:** Number Theory

[1545] **viXra:1708.0204 [pdf]**
*submitted on 2017-08-17 05:25:20*

**Authors:** Ramaswamy Krishnan

**Comments:** 2 Pages.

If\quad f(a)\quad =\quad 1\quad -\quad { a }^{ p }\quad -\quad { (1-a) }^{ p }\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad and\quad even\quad if\quad { a }^{ 2 }-a+1\quad \ncong \quad 0\quad mod(p)\\ it\quad is\quad prooved\quad that\quad f({ a }_{ r })\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad .Then\quad using\quad the\quad fact\quad that\quad if\\ { 3 }^{ p-1 }\quad \equiv \quad 1\quad mod({ p }^{ 3 })\quad ,\quad { a }^{ 2 }+a+1\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad is\quad also\quad a\quad solution\quad to\quad \\ f(a)\quad \equiv \quad 0\quad mod({ p }^{ 3 }).

**Category:** Number Theory

[1544] **viXra:1708.0187 [pdf]**
*submitted on 2017-08-16 12:56:13*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

This note presents some formulas for pi.

**Category:** Number Theory

[1543] **viXra:1708.0181 [pdf]**
*submitted on 2017-08-16 06:53:04*

**Authors:** Kurmet Sultan

**Comments:** 29 pages, Written in Russian

The article presents the proof of the lonely runner conjecture.

**Category:** Number Theory

[1542] **viXra:1708.0177 [pdf]**
*submitted on 2017-08-16 00:48:13*

**Authors:** Kurmet Sultan

**Comments:** 49 Pages.

The article provides with the evidence of the Collatz conjecture.

**Category:** Number Theory

[1541] **viXra:1708.0158 [pdf]**
*submitted on 2017-08-14 08:00:35*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 3 Pages.

I found a pattern in the Pythagorean triples formed of the natural number12;{(12,5,13), (12,9,15), (12,16, 20), (12,35,37)}. The pattern is the decreasing value of the difference z - y for the triples such that the differences form a sequence of
the even numbers {8,6,4,2} in that order. The existence of such sequence for other natural numbers transforms the Pythagorean equation into a linear
equation in y .

**Category:** Number Theory

[1540] **viXra:1708.0141 [pdf]**
*submitted on 2017-08-13 04:27:47*

**Authors:** Hervé G.

**Comments:** 4 Pages.

Yet another proof that zeta(2)=Pi^2/6

**Category:** Number Theory

[1539] **viXra:1708.0114 [pdf]**
*submitted on 2017-08-10 21:43:45*

**Authors:** Joseph Dise

**Comments:** 3 Pages.

A minimum number of paired composite sums is shown for all 2N. By logical extension, it proves the existence of paired prime sums for all 2N.

**Category:** Number Theory

[1538] **viXra:1708.0108 [pdf]**
*submitted on 2017-08-10 09:44:07*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The P-alic numbers are defined and we show a formula giving the decomposition of a polynomial according the valuations.

**Category:** Number Theory

[1537] **viXra:1708.0103 [pdf]**
*submitted on 2017-08-09 13:24:14*

**Authors:** Joseph Dise

**Comments:** 1 Page.

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.

**Category:** Number Theory

[1536] **viXra:1708.0082 [pdf]**
*submitted on 2017-08-08 05:09:33*

**Authors:** François Mendzina Essomba

**Comments:** 12 Pages.

I propose new formulas for the transcendent functions that I discovered.

**Category:** Number Theory

[1535] **viXra:1708.0076 [pdf]**
*submitted on 2017-08-08 07:03:16*

**Authors:** Zhang Tianshu

**Comments:** 22 Pages.

Positive integers which are able to be operated to 1 by set operational rule of the Collatz conjecture and positive integers got via operations by the operational rule versus the set operational rule are one-to-one the same, thus we refer to converse operational routes, apply the mathematical induction, next classify positive integers to prove the Collatz conjecture by substeps according to beforehand prepared two theorems concerned.

**Category:** Number Theory

[1534] **viXra:1708.0063 [pdf]**
*submitted on 2017-08-06 18:22:13*

**Authors:** Mendzina Essomba Francois

**Comments:** 1 Page.

A strange formula for the calculation of the natural logarithm of any real number which I guessed almost effortlessly.
This formula has the advantage of being very efficient for the calculation of the logarithm of large numbers

**Category:** Number Theory

[1533] **viXra:1708.0048 [pdf]**
*submitted on 2017-08-04 15:49:56*

**Authors:** Mendzina Essomba Francois

**Comments:** 3 Pages.

I present some formulas found for the calculation of phy. Some of these formulas accelerate the calculation of decimals and others are written as a function of e and pi by continuous fractions

**Category:** Number Theory

[1532] **viXra:1708.0046 [pdf]**
*submitted on 2017-08-05 03:08:52*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

some of my many pi formulae

**Category:** Number Theory

[1531] **viXra:1708.0044 [pdf]**
*submitted on 2017-08-04 11:37:00*

**Authors:** Antoine Balan

**Comments:** 6 pages, written in french

Foundations of numbers theory are reviewed and some basic definitions are studied.

**Category:** Number Theory

[1530] **viXra:1707.0410 [pdf]**
*submitted on 2017-07-31 10:45:34*

**Authors:** Victor Sorokine

**Comments:** 6 Pages. English version

The essence of the proof:
From the known properties of the Fermat’s equality An+Bn=Cn follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /stay/ infinitely large.

**Category:** Number Theory

[1529] **viXra:1707.0395 [pdf]**
*submitted on 2017-07-29 16:17:02*

**Authors:** Ramón Ruiz

**Comments:** 36 Pages. This document is written in Spanish

Goldbach's Conjecture: “Every even integer greater than 2 can be expressed as the sum of two primes”.
In this document I used the prime numbers theorem enunciated by Carl Friedrich Gauss and the prime numbers theorem in arithmetic progressions. These two theorems applied to a combination of two arithmetic progressions of module 30 and that contain prime numbers, allows us to develop a nonprobability general formula to calculate, approximately, the number of prime pairs that adding up an even number x.
This research is based on a approach designed solely to demonstrate the Binary Goldbach Conjecture and the Twin Prime Conjecture.

**Category:** Number Theory

[1528] **viXra:1707.0392 [pdf]**
*submitted on 2017-07-30 03:32:36*

**Authors:** L. Castillo

**Comments:** 9 Pages.

I explore a method to characterize all the real numbers a,b such that all of $a - b, a^2 - b^2,...,a^n - b^n$ are integers for a given n and paying particular attention to the special case when neither of a and b are integers themselves.

**Category:** Number Theory

[1527] **viXra:1707.0335 [pdf]**
*submitted on 2017-07-26 03:43:24*

**Authors:** Idriss Olivier Bado

**Comments:** In 4 pages i give the proof to that conjecture

In this paper we give the proof of even gap conjecture whose can be expressed by it exists infinitely prime p such that p+n is prime for an even integer n and we deduce Polignac conjecture

**Category:** Number Theory

[1526] **viXra:1707.0279 [pdf]**
*submitted on 2017-07-20 13:09:45*

**Authors:** Emmanuil Manousos

**Comments:** 29 Pages.

Natural numbers have a strictly defined internal structure that is being revealed in the present article. This structure is inherent of the natural numbers and is not derived through the introduction of any axioms for the set of natural numbers. In the present article, we prove the fundamental theorems that determine this structure. As a consequence of this structure, a mathematical expression for the set of odd numbers that are not primes is derived. Given the set of odd numbers, we can identify the set of prime numbers. Additionally, a new method for expressing odd composite numbers as the product of powers of prime numbers is derived.

**Category:** Number Theory

[1525] **viXra:1707.0258 [pdf]**
*submitted on 2017-07-18 15:08:59*

**Authors:** Rédoane Daoudi

**Comments:** 5 Pages.

In this short paper we propose a new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞ .

**Category:** Number Theory

[1524] **viXra:1707.0241 [pdf]**
*submitted on 2017-07-17 13:26:16*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

This note presents formulas and fractals related with Ramanujan's trigonometric formula.

**Category:** Number Theory

[1523] **viXra:1707.0240 [pdf]**
*submitted on 2017-07-17 14:34:07*

**Authors:** François Mendzina Essomba

**Comments:** 5 Pages.

I present in this article some of my many formulas discovered on pi

**Category:** Number Theory

[1522] **viXra:1707.0237 [pdf]**
*submitted on 2017-07-17 22:43:49*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have found the possible max- difference between two successive prime numbers, and by them, Lengendre's conjecture is verified.

**Category:** Number Theory

[1521] **viXra:1707.0217 [pdf]**
*submitted on 2017-07-15 15:40:09*

**Authors:** Mendzina Essomba Francois

**Comments:** 1 Page.

How to prove that an integer number is prime with the factoriels.
We give in this article which is not complete a property of the facoral which allows in an interval of given length to verify if the number is prime

**Category:** Number Theory

[1520] **viXra:1707.0176 [pdf]**
*submitted on 2017-07-13 03:54:39*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[1519] **viXra:1707.0174 [pdf]**
*submitted on 2017-07-12 07:32:30*

**Authors:** Victor Sorokine

**Comments:** 4 Pages.

The proof of Fermat's last theorem for the base case /
Доказательство ВТФ для базового случая
ABSTRACT
The essence of the proof:
From the known properties of the Fermat’s equality An+Bn=Cn follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /remain/ infinitely large.
Суть доказательства:
Из базового равенства Ферма An+Bn=Cn следует:
Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля, то новые уменьшенные числа А°, В°, С° становятся /остаются/ бесконечно большими.
(See also http://vixra.org/abs/1707.0092)

**Category:** Number Theory

[1518] **viXra:1707.0168 [pdf]**
*submitted on 2017-07-11 17:00:37*

**Authors:** Wes Hansen

**Comments:** 100 Pages.

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.

**Category:** Number Theory

[1517] **viXra:1707.0167 [pdf]**
*submitted on 2017-07-11 18:24:29*

**Authors:** Leszek Włodzimierz Guła

**Comments:** 7 Pages. In this work we have a new deductions.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture.

**Category:** Number Theory

[1516] **viXra:1707.0152 [pdf]**
*submitted on 2017-07-10 13:09:21*

**Authors:** Rédoane Daoudi

**Comments:** 5 Pages.

In this paper we propose a conjecture about prime numbers. Based on the result of Pierre Dusart stating that the n th prime number is smaller than n(ln n + ln ln n − 0.9484) for n ≥ 39017 we propose that the n th prime number is smaller than n(ln n + ln ln n − 1+) when n → +∞.

**Category:** Number Theory

[1515] **viXra:1707.0092 [pdf]**
*submitted on 2017-07-06 04:02:34*

**Authors:** Victor Sorokine

**Comments:** 6 Pages. The text is in French

English. The essence of the proof
From the known properties of the Fermat’s equality A n +B n =C n follows:
If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero,
then the new reduced numbers A°, B°, C° become infinitely large.
From which follows the truth of the FLT?
Français. L'essence de la preuve :
A partir des propriétés connues de l'égalité de Fermat A n +B n =C n il suit:
Si les deuxièmes chiffres de tous les facteurs premiers des nombres A, B, C, réduit à zéro,
alors les nombres de nouveaux A°, B°, C°, devenir infiniment grand.
Ce qui implique la vérité du DTF ?
Русский. Суть доказательства:
Из базового равенства Ферма A n +B n =C n следует:
Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля,
то новые уменьшенные числа А°, В°, С° становятся бесконечно большими.
Из чего следует истинность ВТФ?

**Category:** Number Theory

[1514] **viXra:1707.0086 [pdf]**
*submitted on 2017-07-05 14:31:20*

**Authors:** François Mendzina Essomba

**Comments:** 5 Pages.

I have come to the conclusion, after finishing a first reection on infinite sums, that all the functions which are written in the form of an infinite sum are written according to the famous Zeta function, this statement is explicitly presented in this article.

**Category:** Number Theory

[1513] **viXra:1707.0048 [pdf]**
*submitted on 2017-07-05 03:01:29*

**Authors:** Muneer Jebreel Karama

**Comments:** 2 Pages.

A positive integer is called Fixed Happy Cube Numbers (FHCN) in case, if you are cubing its digits and adding them together one time you got the same number. For example the number 153 is happy cube because;
153= 1^3+5^3+3^3, in fact this paper will address new propriety of this extraordinary happy cube number .

**Category:** Number Theory

[693] **viXra:1711.0296 [pdf]**
*replaced on 2017-11-17 23:40:31*

**Authors:** Kurmet Sultan

**Comments:** 9 Pages. Russian version

In this paper we give a brief proof of the Collatz conjecture. It is shown that it is more efficient to start calculating the Collatz function C (n) from odd numbers 6m ± 1. It is further proved that if we calculate by the formula ((6n ± 1)·2^q -1) / 3 on the basis of a sequence of numbers 6n ± 1, increasing the exponent of two by 1 at each iteration, then to each number of the form 6n ± 1 there will correspond a set whose elements are numbers of the form 3t, 6m-1 and 6m + 1. Moreover, all sets are disjoint. Then it is shown that if we construct micro graphs of numbers by combining the numbers 6n ± 1 with their elements of the set 3t, 6m-1 and 6m + 1, then combine the micro graphs by combining equal numbers 6n ± 1 and 6m ± 1, then a tree-like fractal graph of numbers. A tree-like fractal graph of numbers, each vertex of which corresponds to numbers of the form 6m ± 1, is a proof of the Collatz conjecture, since any of its vertices is connected with a finite vertex connected with unity.

**Category:** Number Theory

[692] **viXra:1711.0291 [pdf]**
*replaced on 2017-11-16 09:44:04*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. Clarifications of lemmas, slight re-organization.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.

**Category:** Number Theory

[691] **viXra:1711.0291 [pdf]**
*replaced on 2017-11-14 09:52:19*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. Minor clarifications and edits of previous version.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.

**Category:** Number Theory

[690] **viXra:1711.0258 [pdf]**
*replaced on 2017-11-10 12:40:06*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. A more complete bibliography is included.

This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.

**Category:** Number Theory

[689] **viXra:1711.0134 [pdf]**
*replaced on 2017-11-10 10:10:06*

**Authors:** Philip Gibbs, Judson McCranie

**Comments:** 9 Pages.

All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps.

**Category:** Number Theory

[688] **viXra:1711.0130 [pdf]**
*replaced on 2017-11-09 06:45:07*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. Slight corrections.

Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.

**Category:** Number Theory

[687] **viXra:1711.0130 [pdf]**
*replaced on 2017-11-04 11:58:03*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. Suitable for first year, first term calculus students: just derivatives of polynomials.

Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.

**Category:** Number Theory

[686] **viXra:1711.0109 [pdf]**
*replaced on 2017-11-05 03:05:10*

**Authors:** Antoine Balan

**Comments:** 5 Pages.

We introduce a generalization of the q-calculus, which we call qq'-calculus. Some formulas are obtained; however the theory remains limited.

**Category:** Number Theory

[685] **viXra:1711.0109 [pdf]**
*replaced on 2017-11-02 16:01:26*

**Authors:** Antoine Balan

**Comments:** 5 Pages.

We introduce here a generalization of the q-calculus which we call the qq'-calculus. Some limited formulas are obtained like the Taylor's expansion formula.

**Category:** Number Theory

[684] **viXra:1710.0145 [pdf]**
*replaced on 2017-11-02 10:23:05*

**Authors:** Timothy W. Jones

**Comments:** 20 Pages. Some corrections and explanations added.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(n), n>1, is irrational.

**Category:** Number Theory

[683] **viXra:1710.0145 [pdf]**
*replaced on 2017-10-21 05:31:18*

**Authors:** Timothy W. Jones

**Comments:** 19 Pages.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(2n+1), n>1, is irrational.

**Category:** Number Theory

[682] **viXra:1709.0312 [pdf]**
*replaced on 2017-10-27 20:31:16*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[681] **viXra:1709.0312 [pdf]**
*replaced on 2017-10-24 21:31:09*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[680] **viXra:1709.0312 [pdf]**
*replaced on 2017-09-24 03:38:04*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[679] **viXra:1708.0220 [pdf]**
*replaced on 2017-08-23 05:44:34*

**Authors:** Ranganath G. Kulkarni

**Comments:** 2 Pages.

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann hypothesis may be wrong. We need to study the variation of new variable r with the given number x.

**Category:** Number Theory

[678] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-12 18:30:40*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

We define here a p-adic like field, called the P-alic field. We replace in fact the ring Z by Q[X] and we show a formula of decomposition of a polynomial.

**Category:** Number Theory

[677] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-11 13:40:29*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Following the definition of p-adic numbers, we apply the definitions to the ring Z[X] instead of Z, we show a formula for a polynomial.

**Category:** Number Theory

[676] **viXra:1708.0108 [pdf]**
*replaced on 2017-08-10 18:38:28*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The P-alic numbers are defined following the definition of p-adic numbers. We show a formula for a polynomial.

**Category:** Number Theory

[675] **viXra:1708.0103 [pdf]**
*replaced on 2017-09-24 12:48:05*

**Authors:** Joseph Dise

**Comments:** 1 Page.

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.

**Category:** Number Theory

[674] **viXra:1707.0335 [pdf]**
*replaced on 2017-07-26 07:25:36*

**Authors:** Idriss Olivier Bado

**Comments:** In 4 pages i give the proof

In this paper i give the proof of Polignac conjecture and even gap cobjecture by using Chebotarev Artin theorem

**Category:** Number Theory

[673] **viXra:1707.0176 [pdf]**
*replaced on 2017-10-19 18:21:15*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[672] **viXra:1707.0176 [pdf]**
*replaced on 2017-09-08 04:10:25*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.

**Category:** Number Theory

[671] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-21 06:58:15*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

**Category:** Number Theory

[670] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-15 02:12:22*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

**Category:** Number Theory

[669] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-14 06:25:46*

**Authors:** John Atwell Moody

**Comments:** 9 Pages.

**Category:** Number Theory

[668] **viXra:1707.0176 [pdf]**
*replaced on 2017-07-13 07:41:09*

**Authors:** John Atwell Moody

**Comments:** 8 Pages.

**Category:** Number Theory

[667] **viXra:1707.0168 [pdf]**
*replaced on 2017-08-05 15:40:50*

**Authors:** Wes Hansen

**Comments:** 100 Pages.

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.

**Category:** Number Theory

[666] **viXra:1707.0167 [pdf]**
*replaced on 2017-08-03 18:41:24*

**Authors:** Leszek Włodzimierz Guła

**Comments:** 10 Pages. This is the new variant of my work.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture. The proof of the Beal’s Conjecture.

**Category:** Number Theory