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A218802
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Decimal expansion of largest x such that x^2 = Gamma(x+1).
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2
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3, 5, 6, 2, 3, 8, 2, 2, 8, 5, 3, 9, 0, 8, 9, 7, 6, 9, 1, 4, 1, 5, 6, 4, 4, 3, 4, 2, 7, 4, 7, 6, 1, 0, 3, 1, 1, 7, 8, 1, 1, 0, 6, 4, 7, 5, 0, 9, 7, 2, 1, 6, 1, 9, 4, 3, 3, 7, 9, 2, 0, 3, 1, 1, 7, 0, 0, 5, 4, 1, 6, 7, 6, 5, 0, 8, 5, 5, 6, 5, 6, 0, 2, 6, 5, 4, 7, 6, 3, 8, 8, 6, 4, 5, 0, 9, 2, 4, 0, 2, 3, 6, 0, 2, 6, 3, 7
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OFFSET
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1,1
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COMMENTS
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In other words, the largest number whose square and factorial coincide.
As one knows from the famous illustration in Jahnke and Emde (p. 13) (or Abramowitz and Stegun, p. 255), there are infinitely many solutions to x^2 = x!. For example, there is another solution near -1.8065. - N. J. A. Sloane, Dec 24 2012
Decimal expansion of greatest real fixed point of Gamma(x). (The only other positive fixed point is 1.) - Rick L. Shepherd, Feb 24 2014
The interval (c, x) = (0.2541970697269031..., 3.562382285390897691415...) with Gamma(c) = x = Gamma(x) is the interval of convergence to 1 of the iteration x->Gamma(x). - Andrea Pinos, Jul 06 2023
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 255.
Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 13.
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LINKS
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EXAMPLE
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3.562382285390897691415...
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MAPLE
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Digits:= 150:
s:= convert(fsolve(x^2 = GAMMA(x+1), x=7/2)/10, string):
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MATHEMATICA
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RealDigits[x /. FindRoot[x^2 == Gamma[x + 1], {x, 3}, WorkingPrecision -> 100]][[1]] (* Bruno Berselli, Dec 24 2012 *)
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PROG
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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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