Hello, the 2 numbers are in the OEIS database as well as mine here, and nothing else : I tried the 'try hard' function and nothing came out. the 2 entries are A074455 and A074454. I am afraid that these 2 numbers are solution of trans. equations which are in general not related to anything we know apart from Pi and the LambertW function. I tried most of the tools I know on it : nothing. Best regards, Simon Plouffe Le 23/12/2014 22:54, Daniel Asimov a écrit :
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
* let d_Amax := the real dimension d where the formula
A(d) = 2 pi^(d/2) / Gamma(d/2)
for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum.
-and-
* let d_Vmax := the real dimension d where the formula
V(d) = pi^(d/2) / Gamma(d/2 + 1)
for the d-dimensional content of the unit d-ball in R^d takes its maximum
Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.
In particular they have the same fractional part:
upsilon := 0.256946404860576780132838388690769236619+
QUESTION: Is anything known about the number theoretic properties of upsilon?
Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown?
--Dan
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