%I #14 May 28 2021 06:14:18
%S 1,5,2,5,7,9,6,4,5,7,2,0,0,6,7,9,7,7,3,6,0,1,6,3,3,2,7,9,9,6,9,7,9,0,
%T 2,2,4,3,3,0,6,8,7,5,3,4,4,9,2,0,3,6,7,2,5,6,8,7,5,8,1,0,2,3,4,9,2,2,
%U 8,4,0,5,0,9,6,6,2,9,5,8,2,3,3,2,7,2,6,6,9,7,3,6,4,7,7,0,7,4,8,5,5,3,0,8,9,7,3,5,1,2,9,5
%N Decimal expansion of least positive x such that Gamma(x) = x^2.
%C Motivated by a question about solutions to (x1)! = x^2 in a mathematical discussion group. As can be seen from growth of the respective functions, the only solution in the positive integers is x = 1; (51)! ~ 5^2 is a near miss.
%C If (x1)! is replaced by the Gamma function Gamma(x), then in addition to the positive noninteger solution x = 5.0367... (see A264785) there are noninteger solutions increasingly close to each negative integer: x = 1.5259... is the largest one (also the farthest from an integer), then come 1.806544..., 3.017901..., 3.997382..., 5.000333... etc. They are to the left of odd and to the right of even negative integers, and the reciprocals of the distances 1/(x[n]+n), rounded to nearest integers, go: 2, 5, 56, 382, 3002, ... (see A339167).
%e The largest negative solution to Gamma(x) = x^2 is x =
%e 1.525796457200679773601633279969790224330687534492036725687581023492284050966...
%t RealDigits[x /. FindRoot[Gamma[x]  x^2, {x, 3/2}, WorkingPrecision > 120], 10, 100][[1]] (* _Amiram Eldar_, May 28 2021 *)
%o (PARI) localprec(5+N=100);digits(solve(x=1.7,1.1,gamma(x)x^2)\.1^N)
%Y Cf. A264785 (solution near 5.036...), A339167.
%K nonn,cons
%O 1,2
%A _M. F. Hasler_, Nov 25 2020
