A060295 Decimal expansion of exp(Pi*sqrt(163)).

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

Offset

18,1

Comments

From Alexander R. Povolotsky, Jun 23 2009, Apr 04 2012: (Start)

One could observe that the last four of Class Number 1 expressions in T. Piezas "Ramanujan Pages" could be expressed as the following approximation:

exp(Pi*sqrt(19+24*n)) =~ (24*k)^3 + 31*24

which gives 4 (four) "almost integer" solutions:

1) n = 0, 19+24*0 = 19, k = 4;

2) n = 1, 19+24*1 = 43, k = 40;

3) n = 2, 19+24*2 = 67, k = 220;

4) n = 6, 19+24*6 = 163, k = 26680; this of course is the case for Ramanujan constant vs. its integer counterpart approximation. (End)

From Alexander R. Povolotsky, Oct 16 2010, Apr 04 2012: (Start)

Also if one expands the left part above to exp(Pi*sqrt(b(n))) where b(n) = {19, 25, 43, 58, 67, 163, 232, ...} then the expression (exp(Pi*sqrt(b(n))))/m (where m is either integer 1 or 8) yields values being very close to whole integer value:

Note, that the first differences of b(n) are all divisible by 3, giving after the division: {2, 6, 5, 3, 32, 33, ...}. (End)

From Amiram Eldar, Jun 24 2021: (Start)

This constant was discovered by Hermite (1859).

It is sometimes called "Ramanujan's constant" due to an April Fool's joke by Gardner (1975) in which he claimed that Ramanujan conjectured that this constant is an integer, and that a fictitious "John Brillo" of the University of Arizona proved it on May 1974.

In fact, Ramanujan studied similar near-integers of the form exp(Pi*sqrt(k)) (e.g., A169624), but not this constant.

Gauld (1984) discovered that (Pi*sqrt(163))^e = 22806.9992... is also a near-integer. (End)

References

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 106.

Harold M. Stark, An Introduction to Number Theory, Markham, Chicago, 1970, p. 179.

Dimitris Vathis, Letter to N. J. A. Sloane, Apr 22 1985.

Links

Harry J. Smith, Table of n, a(n) for n = 18..20000

Jens Blanck, Exact real arithmetic systems: results of competition, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science 2064/2001.

Richard E. Borcherds, MegaFavNumbers 262537412680768000, video (2020).

R. F. Churchhouse and S. T. E. Muir, Continued fractions, algebraic numbers and modular invariants, IMA Journal of Applied Mathematics, Vol. 5, No. 3 (1969), pp. 318-328; alternative link.

Alex Clark and Brady Haran, 163 and Ramanujan Constant, Numberphile video (2012).

Philip J. Davis, Are there coincidences in mathematics?, The American Mathematical Monthly, Vol. 88, No. 5 (1981), pp. 311-320.

Martin Gardner, Six Sensational Discoveries That Somehow or Another Have Escaped Public Attention, Mathematical Games, Scientific American, Vol. 232, No. 4 (1975), pp. 126-133.

David Barry Gauld, Problem 12 revisited, New Zealand Mathematical Society Newsletter 32 (December 1984), p. 17.

I. J. Good, What is the Most Amazing Approximate Integer in the Universe?, Pi Mu Epsilon Journal, Vol. 5, No. 7 (1972),pp. 314-315; entire issue.

Charles Hermite, Sur la théorie des équations modulaires et la résolution de l'équation du cinquième degré, Paris: Mallet-Bachelier, 1859, see p. 48.

D. H. Lehmer, Table to many places of decimals, Queries-Replies, Math. Comp., Vol. 1, No. 1 (1943), pp. 30-31.

Tito Piezas III The Ramanujan pages, see section 05.

Simon Plouffe, exp(pi*sqrt(163)) to 5000 digits.

Simon Plouffe, exp(Pi*sqrt(163)), the Ramanujan number, to a precision of 2000 digits.

C. Radoux, A Formula of Ramanujan(Text in French).

C. Radoux, A Formula of Ramanujan(Continued) (Text in French).

Eric Weisstein's World of Mathematics, Ramanujan Constant.

Index entries for transcendental numbers

Formula

exp(Pi*sqrt(163)) = A199743(6)^3 + 744 - 7.4992... * 10^-13. - Charles R Greathouse IV, Jul 15 2020

Example

The Ramanujan number = 262537412640768743.99999999999925007259719818568887935...

Mathematica

RealDigits[N[E^(Pi*Sqrt[163]), 110]][[1]]

Prog

(PARI) default(realprecision, 20080); x=exp(Pi*sqrt(163))/10^17; for (n=18, 20000, d=floor(x); x=(x-d)*10; write("b060295.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009
(Magma) R:= RealField(); Exp(Pi(R)*Sqrt(163)) // G. C. Greubel, Feb 15 2018

Crossrefs

Cf. A058292, A019297, A093436, A102912, A169624, A181045, A181165, A181166.

Sequence in context: A220279 A221188 A220532 * A102912 A064850 A151853

Adjacent sequences: A060292 A060293 A060294 * A060296 A060297 A060298

Keyword

nonn,easy,cons

Author

Jason Earls, Mar 24 2001

Status

approved