Related papers: A New Way to Proof 3x+1 Problem

In 1937, Lothar Collatz conjectured that the sequence generated by the rule $f(n)=3n+1$ for $n\in\mathbb{N}$ odd, $f(n)=n/2$ for $n\in\mathbb{N}$ even, starting in any positive integer $n$ produces $1$. This is equivalent to (1) there are…

We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…

This paper presents a short and simple proof of the Four-Color Theorem that can be utterly checkable by human mathematicians, without computer assistance. The new key idea that has allowed it and the global structure of the proof are…

Exploring the Collatz Conjecture and changing the expression from 3n + 1 to 5n + 1, we found patterns in different sets of numbers. Some numbers reduce to one (as stated in the Collatz Conjecture), some might escape to infinity, and some…

We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$ for nonnegative…

We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0,…

Let $p$ be a prime greater than 3. In the paper we mainly determine $\sum_{k=0}^{[p/4]}\binom{4k}{2k}(-1)^k$, $\sum_{k=0}^{[p/3]}\binom{3k}k, \sum_{k=0}^{[p/3]}\binom{3k}k(-1)^k$ and $\sum_{k=0}^{[p/3]}\binom{3k}k(-3)^k$ modulo $p$, where…

A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the…

Let $F$ be a quadratic form in four variables, let $m\in\mathbb{N}$ and let $\mathbf{k}\in \mathbb{Z}^4$. We count integer solutions to $F(\mathbf{x})=0$ with $\mathbf{x}\equiv \mathbf{k}\:\mathrm{mod}(m)$. One can compare this to the…

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3.…

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq…

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that…

Let $H$ be a 2-regular graph and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. The "cycles plus $K_4$'s" problem is to show that $G$ is 4-colourable; this is a special case of the \emph{Strong Colouring…

We use stack words to find a new, simple proof for the best known upper bound for the number of 3-stack sortable permutations of a given length. This is the first time that stack words are used to obtain such a result.

An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.

In this paper, we give a new method answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the equation $x^d+(x+1)^d=b$ in $\mathbb{F}_{q^4}$, where $n$ is a positive integer, $q=2^n$ and…

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime…