Linear combinations of prime powers in sums of terms of binary recurrence sequences

Let (U _n ) _n≥0 be a nondegenerate binary recurrence sequence with positive discriminant. Let p ₁ , . . . , p _s be fixed prime numbers, b ₁ , . . . , b _s be fixed nonnegative integers, and a ₁ , . . . , a _t be positive integers. In this paper, under certain assumptions, we obtain a finiteness result for the solution of the Diophantine equation \( {\alpha}_1{U}_{n1}+\cdots +{\alpha}_t{U}_{n1}={b}_1{p}_1^{z_1}+\cdots {b}_s{p}_s^{z_s}. \) Moreover, we explicitly solve the equation F _n1 + F _n2 = 2 ^z1 + 3 ^z2 in nonnegative integers n ₁, n ₂, z ₁, z ₂ with z ₂ ≥ z ₁. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker–Davenport reduction method. This work generalizes the recent papers [E. Mazumdar and S.S. Rout, Prime powers in sums of terms of binary recurrence sequences, arXiv:1610.02774] and [C. Bertók, L. Hajdu, I. Pink, and Z. Rábai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261–271, 2017].     

Correction to: X-coordinates of Pell equations as sums of two Tribonacci numbers

Periodica Mathematica Hungarica - In this work, we correct an oversight from [1].     

A pair of equations in four prime squares and powers of 2

The Ramanujan Journal - In this paper, we obtain when $$k=98$$ , every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of four prime...     

Sums of prime powers

It is shown that iff is a suitable polynomial then every sufficiently large positive integer is a sum of distinct summands taken from the sequence {f(p)}, wherep runs through rational primes.     

三个或多个素变数的线性方程

推广了J.B.Friedlander和D.A.Goldston的结果,给出了素变数整系数线性方程a1p1 a2p2 … akpk=N(k≥3)解的个数的渐近公式.     

Solving constrained Pell equations

Consider the system of Diophantine equations , , where is a given integer polynomial. Historically, such systems have been analyzed by using Baker's method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases and , which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.     

Combinatorial congruences modulo prime powers

Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a -adic order bound given by the authors in a previous paper can be attained when .

     

Powers in recurrence sequences: Pell equations

In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer , we show how to (practically) solve Diophantine equations of the form

in integers and . Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.

     

Universal Kummer congruences mod prime powers

We have previously proved Kummer congruences mod primes p such that for the universal divided Bernoulli numbers . In this paper we strengthen these congruences to hold mod powers of p.     

Links of symbolic powers of prime ideals

In this paper, we prove the following. Let be a Cohen-Macaulay local ring of dimension d ≥ 2. Suppose that either R is not regular or if R is regular assume that d ≥ 3. Let t ≥ 2 be a positive integer. If is a regular sequence contained in , then

On a Diophantine inequality involving prime powers

Suppose that N is a sufficiently large real number. In this paper it is proved that if 1 < c < 108/53, c ≠ 2, then the Diophantine inequality ${| p^c_1 + p^c_2+ p^c_3 +p^c_4+ p^c_5 - N| < \log^{-1}N}$ is solvable in primes p ₁, p ₂, p ₃, p ₄, p ₅. This result constitutes an improvement upon that of Zhai and Cao for the range 1 < c < 81/40, c ≠ 2.     

Varieties of lattice-ordered groups in which prime powers commute

We show that the set of all varieties of lattice ordered groups contained in the metabelian variety defined by the lawx ^p y ^p =y ^p x ^p (wherep is a prime) is a well ordered tower. We give an explicit construction of the subdirectly irreducible members of each of these varieties and show that each variety is defined by a single equation.Presented by L. Fuchs.     

Optimal coefficients modulo prime powers in the three-dimensional case

Summary For the three-dimensional case the existence of optimal coefficients, satisfying additional properties important in the theory of linear congruential pseudorandom numbers, is shown for prime power moduli.     

On an average ternary problem with prime powers

The Ramanujan Journal - We continue our work on averages for ternary additive problems with powers of prime numbers in Languasco and Zaccagnini (J Number Theory 159:45–58, 2016; Rocky...