A181896 Least value x solving x^2 - y^2 = n!

5, 11, 27, 71, 201, 603, 1905, 6318, 21888, 78912, 295260, 1143536, 4574144, 18859680, 80014848, 348776640, 1559776320, 7147792848, 33526120320, 160785625902, 787685472000, 3938427360000, 20082117976800, 104349745817240, 552166953609600, 2973510046027938

Offset: 4

Author

Artur Jasinski, Mar 31 2012

Keywords

• nonn

Comments

Many of terms in this sequence are that same as A055228(n) but not all.

a(n) solves the Brocard-Ramanujan Problem, n! = a(n)^2 - 1, and thus (n, a(n)) are a pair of Brown Numbers, if and only if A038202(n) = 1. - Austin Hinkel, Dec 28 2022

Links

• Eric Weisstein's World of Mathematics, Brocard's Problem.

Crossrefs

For least y value see A038202.

Cf. A055228.

Programs

Mathematica

cc = {}; Do[f = n!/4; x = Max[Select[Divisors[f], # <= Sqrt[f] &]]; kk = f/x - x; AppendTo[cc, Sqrt[n! + kk^2]], {n, 4, 30}]; cc

PARI

a(n)=my(N=n!,x=sqrtint(N));while(!issquare(x++^2-N),);x \\ Charles R Greathouse IV, Apr 10 2012