Related papers: 6m Theorem for Prime numbers

For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1) $ permutation polyomials, all giving involutory permutations with exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials,…

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

We show that for every $0 < \epsilon \leq 1$ and integer $k\geq 1$, there exists an integer $n = n(\epsilon,k)$ so that for all primes $p$, and integers $0 \leq a \leq p-1$, there exist integers $1 \leq x_1 < ... < x_n \leq p^\epsilon$ such…

Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's…

Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n)…

Let $g$ be a map from the set of positive integers into itself defined as follows: Let $x$ be a positive integer. If $x$ is odd, then $g(x)=3x+1$, and if $x$ is even, then $g(x)=x/2$. The $3x+1$ conjecture, also called the Collatz…

For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation…

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$…

In this paper, we provide a generalization of Proth's theorem for integers of the form $Kp^n+1$. In particular, a primality test that requires only one modular exponentiation similar to that of Fermat's test without the computation of any…

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…

The number of primes of a kind x^2+1 is infinite.

Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares…

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this…

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$…

Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $\nu_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $\nu_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at…

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…