We present simple proofs of Walter Feit’s results on large Zsigmondy primes. We present simple proofs of known results related to Zsigmondy primes. We recall that if a, n are integers greater than 1, then a prime p is called a Zsigmondy prime for 〈a, n〉 if p a and the order of a (mod p) equals n (see [2], [4, §5], and Theorem 3 below). If p is a Zsigmondy prime for 〈a, n〉, then n | p − 1; thus p ≥ n + 1. A Zsigmondy prime p for 〈a, n〉 is called a large Zsigmondy prime for 〈a, n〉 if p > n+1 or p divides an−1 (equivalently, a prime p is a large Zsigmondy prime for 〈a, n〉 iff p is a Zsigmondy prime for 〈a, n〉 satisfying |an − 1|p > n + 1. See [2]). Zsigmondy primes are used in finite group theory (see, e.g., [1]). For applications of large Zsigmondy primes to finite groups see [3] and [4]. The main results that we reprove here are Theorem 3 (Zsigmondy’s Theorem) and Theorem 10 (due to Walter Feit). We now recall some basic properties of cyclotomic polynomials, which we will use below (see, e.g., [6, Preliminaries, sec. 1]). For n ≥ 1 the cyclotomic polynomial Φn(X) is defined as

On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, II

On Zsigmondy primes

Let u
 n
 be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of u
 n
 which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

background or methods

/pdf/on-divisors-of-lucas-and-lehmer-numbers-13jch74yag.pdf

On divisors of Lucas and Lehmer numbers

Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

In 1934 Mahler [7], [9] showed, by a /?-adic generalisation of the Thue-Siegel theorem, that the greatest prime factor of wn, the /?-th term of a non-degenerate binary recurrence sequence, tends to infmity with n. However, because of the ineffective nature of the Thue-Siegel-Roth theorem, Mahler' s proof does not yield an effective lower bound, which tends to infmity with n, for the greatest prime factor of un. In 1967 Schinzel [16], by employing a /7-adic theorem of Gelfond in place of the p-adic ThueSiegel Theorem used by Mahler, was able to give such a lower bound. For any integer m let P (m) denote the greatest prime factor of m with the convention that P(0) = P(±1) = 1. Schinzel proved that (2) P(iO>Oi(log/ir, where C=C(r, s, u0, u{), c\\ and c2 are effectively computable positive numbers; indeed we may take ci =— and c2 = — if α and β are integers, c\\ = —— and ^2 = 7^ otherwise.

background or result

On divisors of terms of linear recurrence sequences.

Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus Dr. Contents 1 Introduction 1 2 Runge-type Diophantine Equations 11 2. Bibliography 65 Samenvatting 73 Curriculum Vitae 74 vii This thesis contains material from the following papers. Introduction In the thesis we shall solve Diophantine equations effectively by various methods, more precisely by Runge's method, Baker's method and Chabauty's method. To put our results in the proper context we summarize some of the relevant history. A Diophantine equation is an equation of the form f (x 1 , x 2 ,. .. , x n) = 0, where f is a given function and the unknowns x 1 , x 2 ,. .. , x n are required to be rational numbers or to be integers. As a generalisation of the concept one may consider rational or integral solutions over a number field. In the study of Diophantine equations there are some natural questions: • Is the equation solvable? • Is the number of solutions finite or infinite? • Is it possible to determine all solutions? Diophantus was a mathematician who lived in Alexandria around 300 A.D. Six Greek books out of thirteen of Diophantus' Arithmetica have been known for a long time. The most famous Latin translation is due to Bachet in 1621. In 1968 an Arabic manuscript was found in Iran, which is a translation from a Greek text written in Alexandria, but probable it was written by some of Diophantus' commentators. In his works he stated mathematical problems and provided rational solutions. To give an idea of the kind of problems we mention here two of them. The first problem is (problem 20 of book 4) to find four numbers such that the product of any two of them increased by 1 is a perfect square. A set with this property is called a (rational) Diophantine quadruple. The set with this property which Diophantus constructed The second problem is problem 17 of book 6 of the Arabic manuscript of Arithmetica which comes down to find positive rational solutions to y 2 = x 6 + x 2 + 1. Diophantus constructed the solution x = 1 2 , y = 9 8. Fermat's Last Theorem concerns the Diophantine equation x n + y n = z n. Fermat (1601-1665) wrote in the margin of an edition of Diophantus' book …

On Divisors of Fermat, Fibonacci, Lucas and Lehmer Numbers, II

Citations

On Zsigmondy primes

On divisors of Lucas and Lehmer numbers

On divisors of Lucas and Lehmer numbers

On divisors of terms of linear recurrence sequences.

Effective Methods for Diophantine Equations

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