Generalized Fermat, double Fermat and Newton sequences
نویسندگان
چکیده
منابع مشابه
The generalized Fermat equation
This article will be devoted to generalisations of Fermat’s equation x + y = z. Very soon after the Wiles and Taylor proof of Fermat’s Last Theorem, it was wondered what would happen if the exponents in the three term equation would be chosen differently. Or if coefficients other than 1 would be chosen. We discuss the reduction of the resolution of such equations to the determination of rationa...
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A search for prime factors of the generalized Fermat numbers Fn(a, b) = a2 n + b2 n has been carried out for all pairs (a, b) with a, b ≤ 12 and GCD(a, b) = 1. The search limit k on the factors, which all have the form p = k · 2m + 1, was k = 109 for m ≤ 100 and k = 3 · 106 for 101 ≤ m ≤ 1000. Many larger primes of this form have also been tried as factors of Fn(a, b). Several thousand new fact...
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This paper is devoted to the generalized Fermat equation xp + yq = zr , where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infini...
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There are two different definitions of generalized Fermat numbers (GFN), one of which is more general than the other. In [5], Ribenboim defines a generalized Fermat number as a number of the form Fa,n = a n + 1 with a > 2, while Riesel ([6]) further generalizes, defining it to be a number of the form a n + b n . Both definitions generalize the usual Fermat numbers Fn = 2 n + 1. The only known F...
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The purpose of this paper is twofold. In the first section we provide sufficient conditions for the divisibility of the class number of certain algebraic number fields by prime powers and applications thereof. We were initially inspired in this regard by Watabe [ 111, wherein some of the results are false thereby leading to the above. In the second section we generalize Fermat’s well-known “two...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2003
ISSN: 0022-314X
DOI: 10.1016/s0022-314x(02)00025-2