[26] **viXra:1706.0543 [pdf]**
*submitted on 2017-06-28 23:55:17*

**Authors:** Liu Ran

**Comments:** 6 Pages.

Thank ancient philosopher Zeno, who brought such an interesting and meaningful paradox. It imply that the limit is reachable. Then we can deduct the infinity is about 618724203×10^26,

**Category:** Number Theory

[25] **viXra:1706.0531 [pdf]**
*submitted on 2017-06-29 05:39:25*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 7 Pages.

In this research investigation, the author has presented a Recursive Past Equation and a Recursive Future Equation based on the Ananda-Damayanthi Normalized Similarity Measure considered to Exhaustion [1] (please see the addendum of [1] as well).

**Category:** Number Theory

[24] **viXra:1706.0507 [pdf]**
*replaced on 2017-06-27 12:34:17*

**Authors:** Thomas Pierre Nicolas Jean Brouard

**Comments:** 3 Pages.

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta function is 1/2. Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.

**Category:** Number Theory

[23] **viXra:1706.0479 [pdf]**
*submitted on 2017-06-25 18:25:43*

**Authors:** Kunle Adegoke

**Comments:** 11 Pages.

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of a certain form.

**Category:** Number Theory

[22] **viXra:1706.0457 [pdf]**
*submitted on 2017-06-23 13:24:49*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we present an integral for the constant pi:pi=3.1415926535...

**Category:** Number Theory

[21] **viXra:1706.0414 [pdf]**
*submitted on 2017-06-21 05:00:47*

**Authors:** Marius Coman

**Comments:** 20 Pages.

A selection of forty sequences regarding primes and Fermat pseudoprimes from my yet unpublished papers, presented in "OEIS style", with definition of the terms of a sequence, examples, few first terms, notes and conjectures.

**Category:** Number Theory

[20] **viXra:1706.0410 [pdf]**
*submitted on 2017-06-21 00:28:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: There exist an infinity of primes p having the property that concatenating s(p) – d(1) with s(p) – d(2) and repeatedly up to s(p) – d(k), where s(p) is the sum of digits of p and d(1),...,d(k) are the digits of p, is obtained a prime q. Example: such prime p is 127 because concatenating 9 (= 10 – 1) with 8 (= 10 – 2) and with 3 (= 10 – 7) is obtained a prime q = 983.

**Category:** Number Theory

[19] **viXra:1706.0408 [pdf]**
*submitted on 2017-06-21 02:28:55*

**Authors:** Choe Ryong Gil

**Comments:** 12 pages, 2 tables

The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known Robin theorem, and prove that the NSC holds for all odd and some even numbers while, the NSC holds for any even numbers under a certain condition, which would be called the condition (d).

**Category:** Number Theory

[18] **viXra:1706.0407 [pdf]**
*submitted on 2017-06-21 02:30:07*

**Authors:** Choe Ryong Gil

**Comments:** 27 pages, 6 tables

In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.

**Category:** Number Theory

[17] **viXra:1706.0381 [pdf]**
*submitted on 2017-06-18 22:43:41*

**Authors:** Lahcen Aghray

**Comments:** 5 Pages.

We obtain a parameterization of a Diophantine equation of degree 4

**Category:** Number Theory

[16] **viXra:1706.0380 [pdf]**
*submitted on 2017-06-18 23:13:05*

**Authors:** Lahcen Aghray

**Comments:** 2 Pages.

The resolution of a Diophantine equation by calculating the intersection of a curve of degree 3 with a line

**Category:** Number Theory

[15] **viXra:1706.0288 [pdf]**
*submitted on 2017-06-15 07:46:58*

**Authors:** Gang Li

**Comments:** 14 Pages.

An attempt of using elementary approach to prove Fermat's last theorem (FLT)
is given. For infinitely many prime numbers, Case I of the FLT can be proved
using this approach. Furthermore, if a conjecture proposed in this paper is
true (k-3 conjecture), then case I of the FLT is proved for all prime numbers.
For case II of the FLT, a constraint for possible solutions is obtained.

**Category:** Number Theory

[14] **viXra:1706.0206 [pdf]**
*submitted on 2017-06-13 13:41:27*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

In this note we recall some formulas related with continued fractions , numbers , sequences and the constant pi.

**Category:** Number Theory

[13] **viXra:1706.0205 [pdf]**
*submitted on 2017-06-13 13:45:55*

**Authors:** Edgar Valdebenito

**Comments:** 10 Pages.

In this note we briefly explore the equation: z^5+z^4-1=0

**Category:** Number Theory

[12] **viXra:1706.0197 [pdf]**
*replaced on 2017-06-28 09:57:27*

**Authors:** Ryan Zielinski

**Comments:** 4 Pages. Version 2 is an extended version of the original paper. Both works are licensed under the CC BY 4.0, a Creative Commons Attribution License.

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.

**Category:** Number Theory

[11] **viXra:1706.0196 [pdf]**
*submitted on 2017-06-14 15:11:32*

**Authors:** Mendzina Essomba Francois

**Comments:** 2 Pages.

J present
two algorithms for calculating the natural logarithm of any real number. The first is an algorithm obtained by the
method of Archimedes for the calculation of pi and the second the product of a succession of rad
icals.

**Category:** Number Theory

[10] **viXra:1706.0192 [pdf]**
*replaced on 2017-07-04 17:51:05*

**Authors:** Leszek W. Guła

**Comments:** 6 Pages.

1. The proper proof of The Fermat's Last Theorem (FLT).
2. 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.

**Category:** Number Theory

[9] **viXra:1706.0134 [pdf]**
*replaced on 2017-09-07 17:27:09*

**Authors:** Kolosov Petro

**Comments:** 13 pages, 5 figures, arXiv:1603.02468, Keywords: Power function, Monomial, Polynomial, Power series, Finite difference, Derivative, Differential calculus, Differentiation, Binomial coefficient, Newton's Binomial Theorem, Exponential function

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown. In Application 3 generalized calculus of finite differences, based on expression (1.9) is shown.

**Category:** Number Theory

[8] **viXra:1706.0112 [pdf]**
*submitted on 2017-06-07 14:51:48*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 6 figures, arXiv:1705.02516

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.
Keywords: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint

**Category:** Number Theory

[7] **viXra:1706.0111 [pdf]**
*submitted on 2017-06-07 19:48:40*

**Authors:** Kolosov Petro

**Comments:** 12 pages, 1 figure, arXiv:1608.00801

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.
Keywords: finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power function, polynomial, monomial, power series, high order derivative, mathematics, differential calculus, math, maths, science, arxiv, preprint, algebra, calculus, open science, differential equations

**Category:** Number Theory

[6] **viXra:1706.0102 [pdf]**
*submitted on 2017-06-06 11:10:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 3-Poulet numbers of the form (6k + 1)*(6h + 1)*(6j + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of pairs of primes [q, r], of the form 6h + 1 and 6j + 1, such that the number obtained concatenating p*q*r – p with p*q*r – q with p*q*r – r then with p*q*r is prime.

**Category:** Number Theory

[5] **viXra:1706.0097 [pdf]**
*submitted on 2017-06-06 04:10:22*

**Authors:** Marius Coman

**Comments:** 2 Pages.

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 2-Poulet numbers of the form (6k + 1)*(6h + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of primes q of the form 6h + 1 such that the number obtained concatenating p*q – p with p*q – q then with p*q is prime.

**Category:** Number Theory

[4] **viXra:1706.0037 [pdf]**
*submitted on 2017-06-05 05:55:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Example: using the sign “//” with the meaning “concatenated to”, for the Poulet number 129921 (= 3*11*31*127), the number (129921 – 3)//(129921 – 11)//(129921 – 31)//(129921 – 127)//129921 = 129918129910129890129794129921 is prime. Note that such primes are obtained for 10 from the first 90 Poulet numbers!

**Category:** Number Theory

[3] **viXra:1706.0033 [pdf]**
*submitted on 2017-06-04 11:53:02*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: for many squares of primes (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(k), where d(1),..., d(k) are the digits of a square of a prime, are primes. Example: for 1369 (= 37^2) the number obtained concatenating 29 = 30 – 1 with 27 = 30 – 3 with 24 = 30 – 6 with 21 = 30 – 9, i.e. the number 29272421, is prime. Note that for 35 from the first 200 squares of primes the numbers obtained this way are primes!

**Category:** Number Theory

[2] **viXra:1706.0032 [pdf]**
*submitted on 2017-06-04 12:51:12*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following observation: for many Poulet numbers (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(n), where d(1),..., d(n) are the digits of a n-digits Poulet number, are primes. Example: for 8911 the number obtained concatenating 22 = 30 – 8 with 21 = 30 – 9 with 29 = 30 – 1 with 29 = 30 – 1, i.e. the number 22212929, is prime.

**Category:** Number Theory

[1] **viXra:1706.0029 [pdf]**
*submitted on 2017-06-04 02:09:22*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 24 Pages.

Introduction of new trigonometric functions and mathematical constants.
The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.

**Category:** Number Theory