[8] **viXra:1309.0154 [pdf]**
*replaced on 2015-07-04 05:41:08*

**Authors:** Golden Gadzirayi Nyambuya

**Comments:** 7 Pages. Proof must now be complete.

English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time -- such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the `truly marvellous' proof, this fact that the proof only came after $358$ years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the `truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it. This proof is so elementary that anyone with a modicum of mathematical prowess in Fermat's days and in the intervening 358 years could have discovered this very proof. This brings us to the tentative conclusion that Fermat might very well have had the `truly marvellous' proof which he claimed to have had and his `truly marvellous' proof may very well have made use of elementary arithmetic methods.

**Category:** Number Theory

[7] **viXra:1309.0145 [pdf]**
*replaced on 2013-12-28 23:17:34*

**Authors:** Marius Coman

**Comments:** 132 Pages.

Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. But we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. Exceptions to Fermat’s “little” Theorem, these numbers seem to be more malleable than primes, more willing to let themselves to be ordered than them, and their depth study will surely shed light on many properties of the primes. I titled the book this way to show how many new and exciting things one can say more about this class of numbers, but, beside the two hundred conjectures (listed in the Part one of this book) and the one hundred and fifty open problems (listed in the Part four of this book), promised in the title, there are many other observations about Fermat pseudoprimes and generic formulas of subsets of Carmichael numbers and Poulet numbers contained in this book. All the articles in this book of collected papers use well-known notions of number theory, with only two exceptions, namely one article in which I defined a new concept, id est “a set of Smarandache-Coman divisors of order k of a composite integer n with m prime factors” and one article in which I have showed several applications of this concept in the study of Fermat pseudoprimes.

**Category:** Number Theory

[6] **viXra:1309.0086 [pdf]**
*submitted on 2013-09-12 22:43:36*

**Authors:** Jinhua Fei

**Comments:** 10 Pages.

In this article，we use the large sieve and the prime number theorem in the arithmetical progression, we gives a nontrivial estimates for the exponential sums with arbitrary bounded coefficients.

**Category:** Number Theory

[5] **viXra:1309.0064 [pdf]**
*submitted on 2013-09-10 01:54:10*

**Authors:** Jinhua Fei

**Comments:** 9 Pages.

In this article, we employ the large sieve and the prime number theorem in the arithmetical progression, we obtain a nontrivial estimate for the exponential sums with arbitrary bounded coefficients.

**Category:** Number Theory

[4] **viXra:1309.0020 [pdf]**
*submitted on 2013-09-05 04:13:20*

**Authors:** Marius Coman

**Comments:** 2 Pages.

To find generic formulas for Poulet numbers (beside, of course, the formula that defines them) was for long time one of my targets; I maybe found such a formula for Poulet numbers with two prime factors, involving the multiples of the number 30, that also is rising an interesting question about primes.

**Category:** Number Theory

[3] **viXra:1309.0012 [pdf]**
*replaced on 2013-09-06 02:26:18*

**Authors:** Zhen Liu

**Comments:** 17 Pages.

Using the method for equation reconstruction of prime sequence, this paper gives the proof that there is at least one prime between positive integers n2 and (n+1)2.The sum and number formulae of primes between any two positive integers are also given out.

**Category:** Number Theory

[2] **viXra:1309.0007 [pdf]**
*submitted on 2013-09-03 07:41:38*

**Authors:** Marius Coman

**Comments:** 2 Pages.

The number 30 is important to me because I always believed in the utility of classification of primes in primes of the form 30k + 1, 30k + 7, 30k + 11, 30k + 13, 30k + 17, 30k + 19, 30k + 23 and 30k + 29 (which may be interpreted as well as primes of the form 30h – 29, 30h – 23, 30h – 19, 30h – 17, 30h – 13, 30h – 11, 30h – 7 and 30h – 1). The following conjecture involves the multiples of the number 30 and is based on the study of Carmichael numbers.

**Category:** Number Theory

[1] **viXra:1309.0005 [pdf]**
*submitted on 2013-09-02 07:31:32*

**Authors:** Marius Coman

**Comments:** 3 Pages.

I was studying the Fermat pseudoprimes in function of the remainder of the division by different numbers, when I noticed that the study of the remainders of the division by 28 seems to be very interesting. Starting from this, I discovered a method to easily find subsequences of Poulet numbers. I understand through “finding subsequences of Poulet numbers” finding such numbers that share a non-trivial property, i.e. not a sequence defined like: “Poulet numbers divisible by 7”.

**Category:** Number Theory