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 A339161 Decimal expansion of least positive x such that Gamma(-x) = x^2. 2

%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 (x-1)! = 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; (5-1)! ~ 5^2 is a near miss.

%C If (x-1)! is replaced by the Gamma function Gamma(x), then in addition to the positive non-integer solution x = 5.0367... (see A264785) there are non-integer 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

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