Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 228.10035
Autor: Erdös, Paul; Ulam, S.
Title: Some probabilistic remarks on Fermat's last theorem. (In English)
Source: Rocky Mountain J. Math. 1, 613-616 (1971).
Review: Define a measure in the space of all sequences of integers. Let the measure of the set of sequences containing n have measure n-\alpha. It is easy to see then that for all sequences neglecting a set of sequences of measure 0, limk = oo ak/k1/(1-\alpha) = c. The authors show that for \alpha > 2/3 with probability one the equation ai+aj = ar has only a boundednumber of solutions but for \alpha \leq 2/3 it has with probability one infinitely many solutions [cf. H.Halberstam and K.F.Roth, Sequences. Vol. I. (1966; Zbl 141.04405)]. Thus speaking very heuristically for k > 3 Fermat's last theorem is true with probability one.
Classif.: * 11D41 Higher degree diophantine equations
11N37 Asymptotic results on arithmetic functions
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