[16] **viXra:1608.0449 [pdf]**
*submitted on 2016-08-31 17:53:09*

**Authors:** Joe Chizmarik

**Comments:** 2 Pages. This is a proof by contradiction.

We first prove a weak form of Fermat's Last Theorem; this unique lemma is key to the entire proof. A corollary and lemma follow inter-relating Pythagorean and Fermat solutions. Finally, we prove Fermat's Last Theorem.

**Category:** Number Theory

[15] **viXra:1608.0439 [pdf]**
*submitted on 2016-08-30 21:46:42*

**Authors:** Watcharakiete Wongcharoenbhorn

**Comments:** 4 Pages. English

We study on the cycle in the Collatz conjecture and there is something surprise us. Our goal is to show that there is no Collatz cycle

**Category:** Number Theory

[14] **viXra:1608.0429 [pdf]**
*replaced on 2017-12-29 05:07:31*

**Authors:** Gyeongmin Yang

**Comments:** 5 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[13] **viXra:1608.0390 [pdf]**
*submitted on 2016-08-28 19:28:48*

**Authors:** Lucas Allen

**Comments:** English, 4 pages, ideas and examples

This paper presents a method of calculating powers and sums of powers using binomial coefficients. The method involves finding analogues of Pascal's triangle for each power and then showing that powers and sums of powers are the sums of binomial coefficients multiplied by constants. The constants are unique for each power. This paper presents a general idea and not a formal proof.

**Category:** Number Theory

[12] **viXra:1608.0356 [pdf]**
*submitted on 2016-08-25 20:25:15*

**Authors:** Zhang Tianshu

**Comments:** 15 Pages.

Positive integers which can operate to 1 by the set operational rule of the conjecture and positive integers got via contrary operations of the set operational rule are one-to-one correspondence unquestionably. In this article, we classify positive integers to prove the Collatz conjecture by the mathematical induction via operations of substep according to confirmed two theorems plus a lemma in advance.

**Category:** Number Theory

[11] **viXra:1608.0144 [pdf]**
*submitted on 2016-08-12 21:14:51*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 06 Pages.

Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r t h order with some new interesting properties.

**Category:** Number Theory

[10] **viXra:1608.0140 [pdf]**
*submitted on 2016-08-12 21:20:08*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, some properties of k Fibonacci and k Lucas numbers are derived and proved by using matrices S and M. The identities we proved are not encountered in the k Fibonacci
and k Lucasnumber literature.

**Category:** Number Theory

[9] **viXra:1608.0139 [pdf]**
*submitted on 2016-08-12 21:22:02*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 04 Pages.

In this paper we defined general matrices Mk(n,m),
Tk,n and Sk(n,m) for k-Fibonacci number. Using these matrices we find some new summation properties for k-Fibonacci and k-Lucas
numbers.

**Category:** Number Theory

[8] **viXra:1608.0138 [pdf]**
*submitted on 2016-08-12 21:23:18*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.

**Category:** Number Theory

[7] **viXra:1608.0137 [pdf]**
*submitted on 2016-08-12 21:24:35*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. There are 32 different schemes of generalization for the Fibonacci sequences of fifth order in the case of two sequences [1]. I introduce their recurrent formulas below.

**Category:** Number Theory

[6] **viXra:1608.0135 [pdf]**
*submitted on 2016-08-12 21:46:46*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence
literature.

**Category:** Number Theory

[5] **viXra:1608.0134 [pdf]**
*submitted on 2016-08-12 21:48:49*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 14 Pages.

In this paper, we defined new relationship between k Fibonacci and k Lucas sequences using continued fractions and series of fractions, this approach is different and never tried in k Fibonacci sequence literature.

**Category:** Number Theory

[4] **viXra:1608.0133 [pdf]**
*submitted on 2016-08-12 21:50:43*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one
sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first
introduced coupled Fibonacci sequences of second order in additive form. There are 8 different schemes of generalization for the Tribonacci sequences in the case of two sequences & there are 16 different schemes of generalization for the Tetranacci sequences in the case of two sequences. I introduce their recurrent formulas below.

**Category:** Number Theory

[3] **viXra:1608.0128 [pdf]**
*submitted on 2016-08-12 13:47:27*

**Authors:** Michael M. Ross

**Comments:** Pages.

By defining a function for a linear equation (of slope-intercept form) to be a composite generator, I am able to show that a subset of odd-value slopes must always have even solutions. I apply this function to generate a parity truth table for any perfect square interval that demonstrates the unequal cardinality of the subsets of odd and even composites. Using elementary set theory I demonstrate that this inequality is deterministic, eliminating the possibility of a prime-free interval. This method successfully attacks Legendre's conjecture, providing a logical-conceptual framework for a formal proof.

**Category:** Number Theory

[2] **viXra:1608.0082 [pdf]**
*submitted on 2016-08-08 11:34:13*

**Authors:** Bengt Månsson

**Comments:** 10 Pages.

A formula giving the $n$:th number of a sequence defined by a recursion formula plus initial value is deduced using generating functions. Of particular interest is the possibility to get an exact expression for the n:th term by means a recursion formula of the same type as the original one. As for the sequence itself it is of some interest that the original recursion is non-linear and the fact that the sequence grows very fast, the number of digits increasing more or less exponentially. Other sequences with the same rekursion span can be treated similarly.

**Category:** Number Theory

[1] **viXra:1608.0062 [pdf]**
*submitted on 2016-08-06 04:06:33*

**Authors:** Lucas Allen

**Comments:** English, 6 pages, equations and examples

This paper presents a “formula” (more or less) for prime numbers in a specific interval. This formula is then used to partially prove the Goldbach conjecture and the twin primes conjecture. The proofs are incomplete however and have not been reviewed by anyone.

**Category:** Number Theory