Variants of the Brocard-Ramanujan equation

Kihel, Omar; Luca, Florian

doi:10.5802/jtnb.631

Cited by 4 publications

(1 citation statement)

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“…The same result holds for values of A which are squares provided that the Szpiro conjecture is true. Some variants on (1.1) were considered by Dąbrowski [7], Kihel and Luca [9] and Kihel et al [10]. One can also consult the paper of Berend and Harmse [1] and the section D25 in Guy [8].…”

Section: Introductionmentioning

confidence: 99%

SOME OBSERVATIONS ON THE DIOPHANTINE EQUATION y²=x!+A AND RELATED RESULTS

Ulas

2012

Bull. Aust. Math. Soc.

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We consider the Brocard-Ramanujan type Diophantine equation y 2 = x! + A and ask about values of A ∈ Z for which there are at least three solutions in the positive integers. In particular, we prove that the set A consisting of integers with this property is infinite. In fact we construct a two-parameter family of integers contained in A. We also give some computational results related to this equation. IntroductionOne among many classical problems in Diophantine equations is a question posed by Brocard in [4,5]. He asked about the characterisation of all integer solutions of the Diophantine equation y 2 = n! + 1.The same question was posed by Ramanujan in [13]. It is well known that (n, y) = (4, 5), (5, 11), (7, 71) are solutions of this equation and that there are no additional solutions with n ≤ 10 9 ; this was proved by Berndt and Galway [2]. Under the assumption of the weak Szpiro conjecture, Overholt [12] proved that there are only finitely many solutions. Let us recall that the Szpiro conjecture says that there exists a constant s > 0 such that, for any triple of positive integers a, b, c with awhere for a given integer m we have N(m) = p|m p, p prime. The number N(m) is called the radical of the integer m and is just the product of primes dividing m

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