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On the diophantine equation \(x!+A=y^ 2\). (English) Zbl 0876.11015

For positive integers \(x,y,A\) (not a square), the diophantine equation in the title has at most finitely many solutions. If \(A\) is square, the finiteness of solutions is implied by a weak form of Szpiro’s conjecture; namely: there exists a constant \(s>0\) such that for any nonzero mutually prime integers \(a,b,c\) with \(a+b=c\), the inequality \(|abc|\leq N_0 (abc)^s\) holds, where \(N_0(n): =\prod_{p|n} p\) denotes the radical of a nonzero integer \(n\) [see S. Lang, Bull. Am. Math. Soc., New Ser. 23, 37-75 (1990; Zbl 0714.11034)].
Reviewer: E.L.Cohen (Ottawa)

MSC:

11D99 Diophantine equations
11A99 Elementary number theory

Citations:

Zbl 0714.11034
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