# Number Theory

## 1608 Submissions

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

### The Proof of Fermat's Last Theorem

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

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

### Cycle and the Collatz Conjecture

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

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

### Expansion of the Euler Zigzag Numbers

Authors: Gyeongmin Yang

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

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

### Using Binomial Coefficients to Find the Sum of Powers

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

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

### A Proof of the Collatz Conjecture (Sixth Revised Version)

Authors: Zhang Tianshu

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

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

### On the Properties of Generalized Multiplicative Coupled Fibonacci Sequence of R T H Order

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

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

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

### On the Properties of K Fibonacci and K Lucas Numbers

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

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

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

### Summation Identities for K-Fibonacci and K-Lucas Numbers Using Matrix Methods

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

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

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

### Fundamental Properties of Multiplicative Coupled Fibonacci Sequences of Fourth Order Under Two Specific Schemes

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

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

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

### Recurrent Formulas of the Generalized Fibonacci Sequences of Fifth Order

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

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 . I introduce their recurrent formulas below.
Category: Number Theory

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

### Determinantal Identities for K Lucas Sequence

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

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

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

### Fibonacci and k Lucas Sequences as Series of Fractions

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

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

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

### Recurrent Formulas of the Generalized Fibonacci Sequences of Third & Fourth Order

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

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

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

### Algorithm for Calculating Terms of a Number Sequence using an Auxiliary Sequence

Authors: Bengt Månsson