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2015 - 1501(14)

Any replacements are listed further down

[869] **viXra:1501.0232 [pdf]**
*submitted on 2015-01-26 17:28:17*

**Authors:** JinHua Fei

**Comments:** 9 Pages.

In this paper, we assume that Hardy-Littlewood Conjecture, we got a better upper bound of the exceptional real zero for a class of module.

**Category:** Number Theory

[868] **viXra:1501.0201 [pdf]**
*submitted on 2015-01-21 17:08:45*

**Authors:** Wu Sheng-Ping

**Comments:** 6 Pages.

The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By analysis in module and a
careful constructing, a condition of non-solution of Diophantine
Equation $a^p+b^p=c^q$ is proved that:
$(a,b)=(b,c)=1,a,b>0,p,q>12$, $p$ is prime. The proof of this
result is mainly in the last two sections.

**Category:** Number Theory

[867] **viXra:1501.0192 [pdf]**
*submitted on 2015-01-20 04:15:40*

**Authors:** Nicolae Bratu

**Comments:** 13 Pages.

This article generalizes and makes some additions to the method used in this demonstration theorem for exponents 3 and 5. In this regard, this paper presents a complete algebraic demonstration of Fermat’s Last Theorem.

**Category:** Number Theory

[866] **viXra:1501.0190 [pdf]**
*submitted on 2015-01-20 00:44:02*

**Authors:** Yu-Lin Chou

**Comments:** 3 Pages.

It is proved that, for every triple of integers $a, b, c \geq 3,$
the Beal equation $x^{a} + y^{b} - z^{c} = 0$ has no solution in
integers
$x, y, z > 0$ such that gcd$(x, y, z) = 1$.
Different than standard Diophantine methods,
our method is to view the expression $x^{a} + y^{b} - z^{c}$ as the
dot product of the linearly independent vectors $(x, y, -z)$ and
$(x^{a-1}, y^{b-1}, z^{c-1})$ and then deduce a contradiction that
the plane through the origin $(0, 0, 0)$ spanned by these two
vectors is not a plane. The proof also shows how the Pythagorean
equation $x^{2} + y^{2} - z^{2} = 0$ does not likewise lead to a
contradiction.

**Category:** Number Theory

[865] **viXra:1501.0150 [pdf]**
*submitted on 2015-01-13 17:48:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I defined the MC(x) function in the following way: Let MC(x) be the function defined on the set of odd positive integers with values in the set of primes such that: MC(x) = 1 for x = 1; MC(x) = x, for x prime; for x composite, MC(x) has the value of the prime which results from the following iterative operation: let x = p(1)*p(2)*...*p(n), where p(1),..., p(n) are its prime factors; let y = p(1) + p(2) +...+ p(n) – (n – 1); if y is a prime, then MC(x) = y; if not, then y = q(1)*q(2)*...*q(m), where q(1),..., q(m) are its prime factors; let z = q(1) + q(2) +...+ q(m) – (m – 1); if z is a prime, then MC(x) = z; if not, it is iterated the operation until a prime is obtained and this is the value of MC(x). In this paper I present a property of this function.

**Category:** Number Theory

[864] **viXra:1501.0146 [pdf]**
*submitted on 2015-01-14 01:42:15*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In two of my previous papers, namely “An interesting property of the primes congruent to 1 mod 45 and an ideea for a function” respectively “On the sum of three consecutive values of the MC function”, I defined the MC function. In this paper I present new interesting properties of three Smarandache type sequences analyzed through the MC function.

**Category:** Number Theory

[863] **viXra:1501.0141 [pdf]**
*submitted on 2015-01-13 05:05:44*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I show a certain property of the primes congruent to 1 mod 45 related to concatenation, namely the following one: concatenating two or three or more of these primes are often obtaied a certain kind of composites, id est composites of the form m*n, where m and n are not necessarily primes, having the property that m + n - 1 is a prime number. Plus, I present an ideea for a function which be interesting to study.

**Category:** Number Theory

[862] **viXra:1501.0129 [pdf]**
*submitted on 2015-01-12 15:53:42*

**Authors:** Ke Xiao

**Comments:** 6 Pages.

Abstract There are many proposed partial prime number formulas, however, no formula can generate all prime numbers. Here we show three formulas which can obtain the entire prime numbers set from the positive integers, based on the Möbius function plus the “omega” function, or the Omega function, or the divisor function.

**Category:** Number Theory

[861] **viXra:1501.0125 [pdf]**
*submitted on 2015-01-12 10:18:33*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

We first get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. After that, expound relations between C and raf (ABC) by the symmetric law of odd numbers. Finally we have proven C≤Cε [raf (ABC)] 1+ ε in which case A+B=C, where gcf (A, B, C) =1.

**Category:** Number Theory

[860] **viXra:1501.0121 [pdf]**
*submitted on 2015-01-11 16:01:11*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present three functions based on the digital sum of a number which might be interesting to study and ten conjectures. These functions are: (I) F(x) defined as the digital sum of the number 2^x – x^2; (II) G(x) equal to F(x) – x and (III) H(x) defined as the digital sum of the number 2^x + x^2.

**Category:** Number Theory

[859] **viXra:1501.0068 [pdf]**
*submitted on 2015-01-05 06:44:12*

**Authors:** Zhang Tianshu

**Comments:** 12 Pages.

We first get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. After that, expound relations between C and raf (ABC) by the symmetric law of odd numbers. Finally we have proven C≤Cε [raf (ABC)] 1+ ε in which case A+B=C, where gcf (A, B, C) =1.

**Category:** Number Theory

[858] **viXra:1501.0067 [pdf]**
*submitted on 2015-01-05 07:14:57*

**Authors:** Zhang Tianshu

**Comments:** 23 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 in which case A, B and C have a common prime factor by concrete examples. After that, prove AX+BY≠CZ in which 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, we have proven that the Beal’s conjecture holds water after the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[857] **viXra:1501.0050 [pdf]**
*submitted on 2015-01-04 03:48:24*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Every odd number y is the sum of n following numbers while n is a divisor of y.

**Category:** Number Theory

[856] **viXra:1501.0026 [pdf]**
*submitted on 2015-01-03 01:06:09*

**Authors:** Jian Ye

**Comments:** 5 Pages.

The Goldbach theorem and the twin prime theorem is homologous.the paper from the prime origin,derived the equations of the twin prime theorem and the Goldbach theorem,and it revealed the equivalence between the Goldbach theorem and the generalized twin prime theorem.

**Category:** Number Theory

[855] **viXra:1412.0273 [pdf]**
*submitted on 2014-12-30 14:23:47*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Collatz conjecture states that starting with any positive integer n, we divide it by 2 if it is even or multiply it by 3 and add 1 if it is odd and repeat this algorithm on the answer always using the same odd or even rule, we will ultimately end up with an answer of 1. Here we prove this conjecture for a special integer which is the product of any prime number “p” greater than three with another positive odd integer “x” that has been derived by using the Fermat’s little theorem and is therefore unique for each prime. Thus we prove Collatz’s conjecture for a small fraction of positive integers “px” which would be expected to roughly represent the same proportion of integers as prime numbers.

**Category:** Number Theory

[854] **viXra:1412.0250 [pdf]**
*submitted on 2014-12-27 04:44:46*

**Authors:** Nicolae Bratu, Adina Cretan

**Comments:** 4 Pages.

The present paper is a fragment revised from the work [3], published only in Romanian. Using a new function, “cubic combination”, we can solve different problems. The novelty of this work consists in the deduction of an infinite number of third degree Ramanujan identities.

**Category:** Number Theory

[853] **viXra:1412.0248 [pdf]**
*submitted on 2014-12-27 05:22:19*

**Authors:** Nicolae Bratu

**Comments:** 8 Pages.

In the work “Disquisitiones Diophanticae”, published in 2006 in Romanian, I had gathered succinctly and schematized the content of the “Memorandum to the Romanian Academy” in 1983, concerning the Fermat’s Last Theorem. This paper demonstrates a lemma representing a completion of the algebraic method proposed by us to prove the Fermat’s Theorem.

**Category:** Number Theory

[852] **viXra:1412.0246 [pdf]**
*submitted on 2014-12-26 16:11:05*

**Authors:** Nicolae BRATU, Adina CRETAN

**Comments:** 6 Pages.

This paper has been updated and completed thanks to suggestions and critics coming from Dr. Mike Hirschhorn, from the University of New South Walles. We want to express our highest gratitude.
The paper appeared in an abbreviated form [6]. The present work is a complete form.
For the homogeneous diophantine equations:x2 + by2 + cz2 = w2 there are solutions in the literature only for particular values of the parameters b and c. These solutions were found by Euler, Carmichael, Mordell. They proposed a particular solution for this equation in [3]. This paper presents the general solution of this equation as functions of the rational parameters b, c and their divisors. As a consequence, we obtain the theorem that every positive integer can be represented as the sum of three squares, with at most one of them duplicated, which improves on the Fermat –Lagrange theorem

**Category:** Number Theory

[851] **viXra:1412.0236 [pdf]**
*submitted on 2014-12-25 04:38:19*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time compositeness test for numbers of the form 2*3^n-1 is introduced .

**Category:** Number Theory

[850] **viXra:1412.0228 [pdf]**
*submitted on 2014-12-24 01:42:47*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make an observation about an interesting formula based on the lesser prime p from a pair of twin primes, id est N = p^3 + 3*p^2 + 4*p + 1, that conducts sometimes to the result N = q*r, where q, r are primes such that r – q + 1 = p and sometimes to the result N = q*r, where at least one from q, r or both are composites such that r – q + 1 = p.

**Category:** Number Theory

[849] **viXra:1412.0223 [pdf]**
*submitted on 2014-12-23 03:31:19*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

A relationship between the harmonic series and the logarithm is presented.

**Category:** Number Theory

[848] **viXra:1412.0212 [pdf]**
*submitted on 2014-12-21 00:16:28*

**Authors:** Carlos Giraldo Ospina

**Comments:** 6 Pages.

In this paper we prove that the equations of Beal's conjecture (BC) and Fermat's last theorem (FLT) are equivalent. This implies that proving FLT means proving BC and vice versa. We give a general proof of FLT for exponents that are multiples of 4. Such proof becomes a general proof for any exponent greater than 2. We also provide a general proof for odd exponents. Each proof is less than one page long, and so simple that any amateur or professional mathematician is able to tell whether it is correct or incorrect.
The first step consists in using the well-known formula for generating Pythagorean triples as well as reductio ad absurdum for exponents that are multiples of 4, and we also use an equivalence relation for any even or odd exponent. The method of reductio ad absurdum is also applied to odd exponents.

**Category:** Number Theory

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

**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

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

**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

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

**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

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

**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

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

**Authors:** Barar Stelian Liviu

**Comments:** 20 Pages.

This paper is protected by copiryght from
16.07.2012
I want to present the paper tu the viXra .

**Category:** Number Theory

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

In a previous paper we derived that if p, p+2 are twin-primes then
2^{p-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

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

**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

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

**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

[404] **viXra:1501.0201 [pdf]**
*replaced on 2015-01-28 20:13:35*

**Authors:** Wu Sheng-Ping

**Comments:** 6 Pages.

The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By analysis in module and a
careful constructing, a condition of non-solution of Diophantine
Equation $a^p+b^p=c^q$ is proved that:
$(a,b)=(b,c)=1,a,b>0,p,q>12$, $p$ is prime. The proof of this
result is mainly in the last two sections.

**Category:** Number Theory

[403] **viXra:1501.0201 [pdf]**
*replaced on 2015-01-28 11:18:10*

**Authors:** Wu Sheng-Ping

**Comments:** 6 Pages.

The main idea of this article is simply calculating integer
functions in module. The algebraic in the integer modules is studied in
completely new style. By analysis in module and a
careful constructing, a condition of non-solution of Diophantine
Equation $a^p+b^p=c^q$ is proved that:
$(a,b)=(b,c)=1,a,b>0,p,q>12$, $p$ is prime. The proof of this
result is mainly in the last two sections.

**Category:** Number Theory

[402] **viXra:1501.0201 [pdf]**
*replaced on 2015-01-24 05:38:48*

**Authors:** Wu Sheng-Ping

**Comments:** 6 Pages.

**Category:** Number Theory

[401] **viXra:1501.0201 [pdf]**
*replaced on 2015-01-23 17:35:12*

**Authors:** Wu Sheng-Ping

**Comments:** 6 Pages.

**Category:** Number Theory

[400] **viXra:1412.0223 [pdf]**
*replaced on 2015-01-02 23:09:53*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

An (assumed) new relationship between the harmonic series $H_{n}$ and the natural logarithm $log(n)$ is presented.

**Category:** Number Theory

[399] **viXra:1412.0223 [pdf]**
*replaced on 2014-12-31 22:00:54*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

An (assumed) new relationship between the harmonic series $H_{n}$ and the natural logarithm $log(n)$ is presented.

**Category:** Number Theory

[398] **viXra:1412.0223 [pdf]**
*replaced on 2014-12-30 01:57:18*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

Numerical evidence for an (assumed) new relationship between the
harmonic series $H_{n}$ and the natural logarithm $log(n)$ is presented. The formula $H_{n}-log(n)$ for the Euler-Mascheroni constant $\gamma$ is adopted accordingly and reveals its (arguable) most elementary form.

**Category:** Number Theory

[397] **viXra:1412.0223 [pdf]**
*replaced on 2014-12-29 03:17:57*

**Authors:** Martin Schlueter

**Comments:** 2 Pages. This document is licensed under a Creative Commons (CC BY-NC-ND)

A relationship between the harmonic series and the logarithm is presented. The formula H(n)-log(n) for the Euler-Mascheroni constant is adopted accordingly.

**Category:** Number Theory

[396] **viXra:1412.0223 [pdf]**
*replaced on 2014-12-28 08:45:38*

**Authors:** Martin Schlueter

**Comments:** 2 Pages. This document is licensed under a Creative Commons (CC BY-NC-ND)

A relationship between the harmonic series and the logarithm is presented. The formula H(n)-log(n) for the Euler-Mascheroni constant is adopted accordingly.

**Category:** Number Theory

[395] **viXra:1412.0223 [pdf]**
*replaced on 2014-12-23 20:42:33*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

A relationship between the harmonic series and the logarithm is presented. The formula H(n)-log(n) for the Euler-Mascheroni constant is adopted accordingly.

**Category:** Number Theory

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

**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