A352548 Decimal expansion of 22*Pi^4.

2, 1, 4, 3, 0, 0, 0, 0, 0, 2, 7, 4, 8, 0, 5, 3, 6, 1, 9, 2, 0, 1, 6, 8, 7, 3, 1, 9, 1, 5, 1, 5, 1, 2, 4, 4, 7, 4, 9, 4, 0, 0, 6, 8, 8, 4, 7, 9, 9, 0, 7, 9, 2, 7, 7, 2, 1, 2, 2, 9, 2, 9, 0, 6, 5, 7, 9, 3, 5, 8, 8, 2, 0, 0, 5, 0, 1, 9, 8, 3, 1, 6, 1, 8, 2, 6, 8, 1, 0, 7, 9, 1, 6, 4

Offset: 4

Author

M. F. Hasler, Jun 21 2022

Keywords

• nonn
• cons
• less

Comments

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 (cf. links) he writes "Divide (...) 2143 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.

Even after a(0..4) = 0, the digits '0' and '1' remain significantly more frequent than other digits: almost 3 times more frequent than the digit 3 within the first 100 terms, and still 30 - 40 percent more frequent than half of the other digits among the first 1000 terms. However, we don't consider that to be a "secret hidden in pi".

References

• Martin Gardner, "Slicing Pi into Millions", Discover, 6:50, January 1985.

Links

• William R. Corliss, The Secret Of It All Is In The Pi, Science Frontiers #37, Jan-Feb 1985.
• Martin Gardner, Slicing Pi into Millions, in: Gardner's Why and Wherefores, Prometheus Books (1999), p. 87.
• Zurab Silagadze, The origin of the Ramanujan's π^4 ≈ 2143/22 identity, MathOverflow.net, Feb 26 2016.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000796 (decimal digits of Pi), A328927 (decimal digits of (2143/22)^1/4).

Programs

Mathematica

RealDigits[22*Pi^4, 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

PARI

A352548_first(N)=localprec(N+5);digits(22*Pi^4\10^(4-N)) \\ First N terms of this sequence, i.e., a(4 .. 5-N).

Formula

22*Pi^4 = 2143.000002748053619201687319151512447494006884799...