Daniele de Rigo

#PiDay

ln( 640320^3 + 744 )/(163)^0.5 = 3.141592653589793238462643383279...

till that last 9 it works, but then it continues with
72661...
rather than 50288... as π does. Nice hoax by Martin Gardner in 1975 [1]

https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan's_constant

#References

[1] Gardner, M., 1975. Mathematical Games - Six sensational discoveries that somehow or another have escaped public attention. Scientific American 232 (4), 126-130. https://doi.org/10.1038/scientificamerican0475-126
(first page freely accessible at https://www.jstor.org/stable/24949779)

Mar 14 at 10:32amWeb
Show threadDaniele de RigoMar 14

2/

Gardner cited #Ramanujan [2] for the general form

[A] : f(n) = exp( π √n )

which is almost integer for some values of n (22, 37, 58, see https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper6/page4.htm). But n=163 was not explicitly mentioned in [2].

#References

[2] Ramanujan Iyengar, S., 1914. Modular equations and approximations to π. The Quarterly Journal of Pure and Applied Mathematics 45, 350-375.
(https://zbmath.org/45.1249.01 - note: eq. [B] here is wrong! ; original:
http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm ; PDF: https://ramanujan.sirinudi.org/Volumes/published/ram06.pdf )

Show threadDaniele de RigoMar 15

3/

Curiously, #Ramanujan [2] noted another approx. [B] (longer) with the same accuracy as in the case [A] for f(163): "31 places of decimals".

In #GNUlinux Bash, easy to compare the two even with bc:

echo 'scale=60; l( 640320^3 + 744 ) / sqrt(163)' | bc -l;

echo 'scale=60; 4/sqrt(522)*l( ((5+sqrt(29))/sqrt(2))^3 * (5*sqrt(29) + 11*sqrt(6) ) * ( sqrt((9+3*sqrt(6))/4 ) + sqrt((5+3*sqrt(6))/4 ) )^6 )' | bc -l;

To show π in bc, use π = 4 arctan(1):

echo 'scale=60; 4*a(1)' | bc -l;

#PiDay