On the equation P(x) = n! and a question of Erdős

Berend, Daniel; Osgood, Charles F.

doi:10.1016/0022-314x(92)90020-p

Cited by 15 publications

(11 citation statements)

References 3 publications

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“…However, his proof for square A still assumes a weaker version of the abc-conjecture. Another generalization was investigated by Berend and Osgood [1], who showed that if P ∈ Z[x] is of degree at least 2, then the density of the set of positive integers n such that (1.2) P (x) = n! has a solution x, is zero.…”

Section: Introductionmentioning

confidence: 99%

A Brocard-Ramanujan-type equation with Lucas and associated Lucas sequences

Pink¹,

Szikszai²

2017

Glas. Mat. Ser. III

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This paper deals with a Brocard-Ramanujan-type equation of the form un 1 un 2 . . . un k + 1 = u 2 m in unknown nonnegative integers k, n 1 , n 2 , . . . , n k and m with k ≥ 1, where u = (un) ∞ n=0 is either a Lucas sequence or its associated sequence. For certain infinite families of sequences we completely solve the above equation, extending some results of Marques [15], Szalay [21] and Pongsriiam [18]. The ingredients of the proofs are factorization properties of Lucas sequences, the celebrated result of Bilu, Hanrot and Voutier on primitive divisors of Lucas sequences and elementary estimations concerning the terms involved.

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Section: Introductionmentioning

confidence: 99%

A Brocard-Ramanujan-type equation with Lucas and associated Lucas sequences

Pink¹,

Szikszai²

2017

Glas. Mat. Ser. III

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“…In particular, the set of n for which equation (10) can have a positive integer solution y is of asymptotic density zero, which is an analogue of the result of Berend and Osgood from [1] for the particular polynomial P (X) = X 2 − 1 and our variant of the Brocard-Ramanujan equation.…”

Section: Theorem 5 If the Weak Form Of Hall's Conjecture Is True Thmentioning

confidence: 72%

“…Brocard (see [4,5]), and independently Ramanujan (see [15,16]), posed the problem of finding all integral solutions to the diophantine equation (1) n! + 1 = x 2 .…”

Section: Introductionmentioning

confidence: 99%

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Variants of the Brocard-Ramanujan equation

Kihel¹,

Luca²

2008

J. Théor. Nombres Bordeaux

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“…has only finitely many integer solutions (n, m) with m positive. Unconditionally, Berend and Osgood [2] showed that the set of positive integers m such that equation (1•3) has an integer solution n is of asymptotic density zero. We refer the reader to [1] for several related results and an extensive bibliography on such problems.…”

Section: Introductionmentioning

confidence: 99%