Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 056.03505
Autor: Ankeny, N.C.; Erdös, Pál
Title: The insolubility of classes of diophantine equations. (In English)
Source: Amer. J. Math. 76, 488-496 (1954).
Review: Let m be a natural and a1,a2,...,an non-zero rational integers such that for every selection ej = 0 or ± 1 (j = 1,2,...,n) except e1 = e2 = ··· = en = 0 we have a1e1+···+anen \ne 0. Let U be a large positive real number tending to infinity and D(U) the number of m \leq U for which the equation a1X1m+a2X2m+···+anXnm = 0 has rational integer solutions in the variables X1,X2,...,Xn where not all Xj = 0. The authors prove that D(U) = o(U). They also prove a case which is excluded in the theorem above: The density of integers m, for which the equations X1m+X2m+X3m = 0 has a rational solution and for which (X1X2X3,m) = 1, is zero. They also mention that the first result can be generalized from the rational number field to any algebraic number field F. The paper contains various misprints and defects. Lemma 4 is incorrect.
Reviewer: S.Selberg
Classif.: * 11D41 Higher degree diophantine equations
14G05 Rationality questions, rational points
Index Words: Number theory
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