

A339161


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


2



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, 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, 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
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OFFSET

1,2


COMMENTS

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.
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).


LINKS



EXAMPLE

The largest negative solution to Gamma(x) = x^2 is x =
1.525796457200679773601633279969790224330687534492036725687581023492284050966...


MATHEMATICA

RealDigits[x /. FindRoot[Gamma[x]  x^2, {x, 3/2}, WorkingPrecision > 120], 10, 100][[1]] (* Amiram Eldar, May 28 2021 *)


PROG

(PARI) localprec(5+N=100); digits(solve(x=1.7, 1.1, gamma(x)x^2)\.1^N)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



