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A328927 Decimal expansion of (9^2 + (19^2)/22)^(1/4): an approximation for Pi from Srinivasa Ramanujan.

%I A328927 #53 Jun 18 2023 01:50:31
%S A328927 3,1,4,1,5,9,2,6,5,2,5,8,2,6,4,6,1,2,5,2,0,6,0,3,7,1,7,9,6,4,4,0,2,2,
%T A328927 3,7,1,5,5,7,8,7,7,9,8,3,1,6,0,1,2,6,1,4,9,6,9,5,1,3,5,3,2,7,9,1,8,6,
%U A328927 2,1,0,5,8,8,4,9,7,8,1,0,1,1,2,3,4,0,8,9,2,6,0,9,5,7,0,3,9,5,5,5
%N A328927 Decimal expansion of (9^2 + (19^2)/22)^(1/4): an approximation for Pi from Srinivasa Ramanujan.
%C A328927 Srinivasa Ramanujan published this curious empirical approximation in 1914 accompanied with a simple geometric construction for Pi based on this value of (9^2 + (19^2)/22)^(1/4) [See Ramanujan link, page 43, section 12, and page 44, Figure 2].
%C A328927 S. Ramanujan found 3.14159265262... as the value for this approximation in 1914 while Maple gives 3.14159265258... and Pi = 3.14159265358...
%C A328927 This approximation is correct to 10^(-8).
%C A328927 Gardner (1985) wrote: "A more astounding discovery is that: 22 pi^4 = 2143. A few multiplications, and the 10 million-plus decimals of pi have vanished. (Can this remarkable relationship mirror some as yet undiscovered facet of physical reality?)" In the Postscript to the 1999 reprint he writes "Divide 2143 (the first four counting numbers) by 22 and hit the square-root button twice. You will get pi to eight decimals", and credits this discovery to Srinivasa Ramanujan. The MathOverflow page also mentions this and the near-integer 10*Pi^4 - 1/11 ≈ 974.0000012... See A352548 for 22*Pi^4. - _M. F. Hasler_, Jun 22 2022
%D A328927 Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved 5 June 2013, (4.18), page 58.
%D A328927 Martin Gardner, "Slicing Pi into Millions", Discover, 6:50, January 1985.
%D A328927 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.
%H A328927 William R. Corliss, <a href="https://www.science-frontiers.com/sf037/sf037p20.htm">The Secret Of It All Is In The Pi</a>, Science Frontiers #37, Jan-Feb 1985.
%H A328927 Martin Gardner, <a href="https://www.yumpu.com/en/document/read/42286079/why-whe/96">Slicing Pi into Millions</a>, in: Gardner's Why and Wherefores, Prometheus Books (1999), p.87.
%H A328927 Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf">Modular equations and approximations to Pi</a>, Quarterly Journal of Mathematics, XLV, 1914, 43-44.
%H A328927 Zurab Silagadze, <a href="https://mathoverflow.net/questions/232177/the-origin-of-the-ramanujans-pi4-approx-2143-22-identity">The origin of the Ramanujan's π^4 ≈ 2143/22 identity</a>, MathOverflow.net, Feb 26 2016.
%H A328927 Wikipedia, <a href="https://en.wikipedia.org/wiki/Approximations_of_%CF%80#Miscellaneous_approximations">Approximations of π: Miscellaneous_approximations</a>.
%F A328927 Equals (102 - 2222/(22^2))^(1/4) = (2143/22)^(1/4).
%e A328927 3.141592652582646125206037179644022371557877983160126149695135327918621058849...
%p A328927 evalf((9^2 + (19^2)/22)^(1/4),125);
%t A328927 RealDigits[Surd[9^2 + (19^2)/22, 4], 10, 120][[1]] (* _Amiram Eldar_, Jun 18 2023 *)
%o A328927 (PARI) A328927_first(N)=localprec(N+9); digits(10^N\sqrtn(22/.2143,4)) \\ First N terms of the sequence, i.e., a(1, 0, -1, ..., 2-N). - _M. F. Hasler_, Jun 22 2022
%Y A328927 Cf. A000796, A068028, A068079, A068089, A352548.
%K A328927 nonn,cons,easy
%O A328927 1,1
%A A328927 _Bernard Schott_, Oct 31 2019