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A339161
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Decimal expansion of least positive x such that Gamma(-x) = x^2.
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2
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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
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1,2
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COMMENTS
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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.
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).
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LINKS
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EXAMPLE
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The largest negative solution to Gamma(x) = x^2 is x =
-1.525796457200679773601633279969790224330687534492036725687581023492284050966...
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MATHEMATICA
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RealDigits[x /. FindRoot[Gamma[-x] - x^2, {x, 3/2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, May 28 2021 *)
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PROG
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(PARI) localprec(5+N=100); digits(solve(x=-1.7, -1.1, gamma(x)-x^2)\.1^N)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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