Primitive Divisors Research Articles

On the exponential Diophantine equation $x^2+p^mq^n=2y^p$

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.

Some effectivity results for primitive divisors of elliptic divisibility sequences

Some effectivity results for primitive divisors of elliptic divisibility sequences

On the Exponential Diophantine Equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$

Let $m$ be a positive integer. In this paper, we consider the exponential Diophantine equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$ and we show that it has only unique positive integer solution $(x,y,z)=(1,1,2)$ for all $ m>1. $ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.

On the exponential Diophantine equation $ (a(a-l)m^{2}+1)^{x}+(alm^{2}-1)^{y} = (am)^{z} $

<abstract><p>Suppose that $ a $, $ l $, $ m $ are positive integers with $ a\equiv1\pmod2 $ and $ a^{2}m^{2}\equiv-2\pmod p $, where $ p $ is a prime factor of $ l $. In this paper, we prove that the title exponential Diophantine equation has only the positive integer solution $ (x, y, z) = (1, 1, 2) $. As an another result, we show that if $ a = l $, then the title equation has positive integer solutions $ (x, y, z) = (n, 1, 2) $, $ n\in\mathbb{N} $. The proof is based on elementary methods, Bilu-Hanrot-Voutier Theorem on primitive divisors of Lehmer numbers, and some results on generalized Ramanujan-Nagell equations.</p></abstract>

On the number of residues of linear recurrences

For every nonconstant monic polynomial \(g \in \mathbb {Z}[X]\), let \(\mathfrak {M}(g)\) be the set of positive integers m for which there exist an integer linear recurrence \((s_n)_{n \ge 0}\) having characteristic polynomial g and a positive integer M such that \((s_n)_{n \ge 0}\) has exactly m distinct residues modulo M. Dubickas and Novikas proved that \(\mathfrak {M}(X^2 - X - 1) = \mathbb {N}\). We study \(\mathfrak {M}(g)\) in the case in which g is divisible by a monic quadratic polynomial \(f \in \mathbb {Z}[X]\) with roots \(\alpha ,\beta \) such that \(\alpha \beta = \pm 1\) and \(\alpha / \beta \) is not a root of unity. We show that this problem is related to the existence of special primitive divisors of certain Lehmer sequences, and we deduce some consequences on \(\mathfrak {M}(g)\). In particular, for \(\alpha \beta = -1\), we prove that \(m \in \mathfrak {M}(g)\) for every integer \(m \ge 7\) with \(m \ne 10\) and \(4 \not \mid m\).

On the Diophantine Equation $$dx^2+p^{2a}q^{2b}=4y^p$$

We investigate the solvability of the Diophantine equation in the title, where \(d>1\) is a square-free integer, p, q are distinct odd primes and x, y, a, b are unknown positive integers with \(\gcd (x,y)=1\). We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.

Some remarks on small values of $$\tau (n)$$

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan $$\tau $$ -function. For example, we prove the existence of an effectively computable positive constant $$\kappa $$ such that if $$\tau (n)$$ is odd and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \; \kappa \cdot \frac{\log \log \log {n}}{\log \log \log \log {n}} \end{aligned}$$ or there exists a prime $$p \mid n$$ with $$\tau (p)=0$$ . Here P(m) denotes the largest prime factor of m. We also solve the equation $$\tau (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$$ and the equations $$\tau (n)=\pm q^b$$ where $$3\le q < 100$$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves.

Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings

In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.

A Lucas\u2013Lehmer approach to generalised Lebesgue\u2013Ramanujan\u2013Nagell equations

We describe a computationally efficient approach to resolving equations of the form C_1x^2 + C_2 = y^n in coprime integers, for fixed values of C_1, C_2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

Perfect powers that are sums of squares of an arithmetic progression

We determine all nontrivial integer solutions to the equation (x+r)2+(x+2r)2+⋯+(x+dr)2=yn for 2≤d≤10 and 1≤r≤104 with gcd(x,y)=1. We make use of a factorization argument and the primitive divisors theorem due to Bilu, Hanrot and Voutier.

SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

AbstractWe find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y&gt;0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For alpha in {text {End}}(E), define B_alpha to be the mathcal {O}_K-integral ideal generated by the denominator of x(alpha (P)+Q). Let mathcal {O} be a subring of {text {End}}(E), that is a Dedekind domain. We will study the sequence {B_alpha }_{alpha in mathcal {O}}. We will show that, for all but finitely many alpha in mathcal {O}, the ideal B_alpha has a primitive divisor when P is a non-torsion point and there exist two endomorphisms gne 0 and f so that f(P)= g(Q). This is a generalization of previous results on elliptic divisibility sequences.

The diophantine equation x2 + paqb = yq

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

On the Diophantine Equation $$cx^2+p^{2m}=4y^n$$

Let c be a square-free positive integer and p a prime satisfying \(p\not \mid c\). Let \(h(-c)\) denote the class number of the imaginary quadratic field \(\mathbb {Q}(\sqrt{-c})\). In this paper, we consider the Diophantine equation $$\begin{aligned}&cx^2+p^{2m}=4y^n,~~x,y\ge 1, m\ge 0, n\ge 3,\\&\quad \gcd (x,y)=1, \gcd (n,2h(-c))=1, \end{aligned}$$and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.

On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors of Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb{Z}[x]$, $a$ is a positive integer and $b$ an arbitrary positive integer.

Primitive divisors of elliptic divisibility sequences for elliptic curves with $j=1728$

Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider the sequence $B_n$ of the denominators of the abscissas of the iterate of a non-torsion point; we show that $B_{5m}$ has a primitive divisor for every $m$. Then, we show how to generalize this method to the terms in the form $B_{mp}$ with $p$ a prime congruent to $1$ modulo $4$.

On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation (x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3=yp only has solutions which satisfy xy=0 for 1≤r≤106 and p≥5 prime. This article complements the work on the equations (x−r)3+x3+(x+r)3=yp in [2] and (x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3=yp in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the j-invariant of the elliptic curve is constant.In more detail, given an elliptic curve E with a point P of infinite order over a global field, the sequence D1, D2,… of denominators of multiples P, 2P,… of P is a strong divisibility sequence in the sense that gcd⁡(Dm,Dn)=Dgcd⁡(m,n). This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences.A number N is called a Zsigmondy bound of the sequence if each term Dn with n>N presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over Q and over function fields.We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is 2 if the j-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.

On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$

Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.