Two $S$-unit equations with many solutions

The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $\epsilon > 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c > (rad(a b c))^{1+\epsilon}$. If true, the abc conjecture would imply many other famous theorems and conjectures as corollaries. In this paper, we discuss the abc conjecture and find new applications to powerful numbers, which are integers ...

Let $(a_1,\dots, a_m)$ be an $m$-tuple of positive, pairwise distinct, integers. If for all $1\leq i< j \leq m$ the prime divisors of $a_ia_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we estimate the number of $S$-Diophantine quadruples in terms of $|S|=r$.

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $\delta x (\log x)^{-1}$ with $\delta\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, \delta x (\log\log x)^{-1}$.

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exc...

We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left [ \frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.

The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many solutions in primes exist.

In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, ...

Let $A\subset [1,x]$ be a set of primes with $|A|= \alpha x (\log x)^{-1}$ and $B\subset [1,x]$ be a set of integers with $|B|= \beta x$. We prove that for any $\kappa>0$ and $\alpha,\beta\in [(\log x)^{-\kappa},1]$, there exists a positive constant $c(\kappa)$ such that $$|A+B|\geq c(\kappa)\frac{\alpha x}{\log 2(\alpha \beta)^{-1}}.$$

We prove that the sumset {p^2+b^2+2^n: p is prime and b,n\in N} has positive lower density. We also construct a residue class with odd modulo, which contains no integer of the form p^2+b^2+2^n. And similar results are established for the sumset {b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in N}.