Related papers: On special Wieferich's primes

This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v,…

Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in…

Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation…

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value…

Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with…

We prove that if $p \equiv 4,7 \pmod{9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb{Q}$.

We prove that on $\mathbb{P}^{3}$ there is no exceptional bundle with rank $r=2d^{2}+1$ and degree $d$ for every $|d|\geq 4$. In particular, we find a new obstruction for the existence of exceptional bundles other than $r|(2d^{2}+1)$. We…

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the…

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.

We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it…

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation $ L_n^{(k)} = L_{n-1}^{(k)} +…

In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order 8p^3, where p \geq 3 is a prime number.

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we generalise this result to \kappa-prime models, for \kappa a regular uncountable cardinal or…

Let b be an odd integer such that b=+/-1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show that if (p(k)) is a sequence of odd primes, with 0<=k<=q-2 such that p(k)=2p(k-1)+b for all 1<=k<=q-2, then…

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that…

In this paper we prove that for any prime $p\ge 11$ holds $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i<j\le p-1}\frac{1}{ij}\pmod{p^7}. $$ This is a generalization of the famous Wolstenholme's theorem which…

We discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$ are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is a positive integer. The technique used to prove Fermat's Last Theorem shows that…

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue…