cp's OEIS Frontend

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

%I A060295 #94 Mar 18 2025 21:40:40
%S A060295 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,
%T A060295 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,
%U A060295 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
%N A060295 Decimal expansion of exp(Pi*sqrt(163)).
%C A060295 From _Alexander R. Povolotsky_, Jun 23 2009, Apr 04 2012: (Start)
%C A060295 One could observe that the last four of Class Number 1 expressions in T. Piezas "Ramanujan Pages" could be expressed as the following approximation:
%C A060295 exp(Pi*sqrt(19+24*n)) =~ (24*k)^3 + 31*24
%C A060295 which gives 4 (four) "almost integer" solutions:
%C A060295 1) n = 0, 19+24*0 = 19, k = 4;
%C A060295 2) n = 1, 19+24*1 = 43, k = 40;
%C A060295 3) n = 2, 19+24*2 = 67, k = 220;
%C A060295 4) n = 6, 19+24*6 = 163, k = 26680; this of course is the case for Ramanujan constant vs. its integer counterpart approximation. (End)
%C A060295 From _Alexander R. Povolotsky_, Oct 16 2010, Apr 04 2012: (Start)
%C A060295 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:
%C A060295 Note, that the first differences of b(n) are all divisible by 3, giving after the division: {2, 6, 5, 3, 32, 33, ...}. (End)
%C A060295 From _Amiram Eldar_, Jun 24 2021: (Start)
%C A060295 This constant was discovered by Hermite (1859).
%C A060295 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.
%C A060295 In fact, Ramanujan studied similar near-integers of the form exp(Pi*sqrt(k)) (e.g., A169624), but not this constant.
%C A060295 Gauld (1984) discovered that (Pi*sqrt(163))^e = 22806.9992... is also a near-integer. (End)
%D A060295 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 225-226.
%D A060295 C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 106.
%D A060295 Harold M. Stark, An Introduction to Number Theory, Markham, Chicago, 1970, p. 179.
%D A060295 Dimitris Vathis, Letter to _N. J. A. Sloane_, Apr 22 1985.
%D A060295 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 142.
%H A060295 Harry J. Smith, <a href="/A060295/b060295.txt">Table of n, a(n) for n = 18..20000</a>
%H A060295 Jens Blanck, <a href="http://dx.doi.org/10.1007/3-540-45335-0_24">Exact real arithmetic systems: results of competition</a>, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science 2064/2001.
%H A060295 Richard E. Borcherds, <a href="https://www.youtube.com/watch?v=a9k_QmZbwX8">MegaFavNumbers 262537412680768000</a>, video (2020).
%H A060295 R. F. Churchhouse and S. T. E. Muir, <a href="https://doi.org/10.1093/imamat/5.3.318">Continued fractions, algebraic numbers and modular invariants</a>, IMA Journal of Applied Mathematics, Vol. 5, No. 3 (1969), pp. 318-328; <a href="https://citeseerx.ist.psu.edu/pdf/aaf7017d3a7ae5ef5e4cbca14df10e2056b30750">CiteSeerX</a>.
%H A060295 Alex Clark and Brady Haran, <a href="https://www.youtube.com/watch?v=DRxAVA6gYMM">163 and Ramanujan Constant</a>, Numberphile video (2012).
%H A060295 Philip J. Davis, <a href="https://www.jstor.org/stable/2320105">Are there coincidences in mathematics?</a>, The American Mathematical Monthly, Vol. 88, No. 5 (1981), pp. 311-320.
%H A060295 Martin Gardner, <a href="https://www.jstor.org/stable/24949779">Six Sensational Discoveries That Somehow or Another Have Escaped Public Attention</a>, Mathematical Games, Scientific American, Vol. 232, No. 4 (1975), pp. 126-133.
%H A060295 David Barry Gauld, <a href="http://nzmathsoc.org.nz/downloads/newsletters/NZMSnews32_Dec1984.pdf">Problem 12 revisited</a>, New Zealand Mathematical Society Newsletter 32 (December 1984), p. 17.
%H A060295 I. J. Good, <a href="https://www.jstor.org/stable/24344898">What is the Most Amazing Approximate Integer in the Universe?</a>, Pi Mu Epsilon Journal, Vol. 5, No. 7 (1972),pp. 314-315; <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.5.No.7.pdf">entire issue</a>.
%H A060295 Charles Hermite, <a href="https://eudml.org/doc/203496">Sur la théorie des équations modulaires et la résolution de l'équation du cinquième degré</a>, Paris: Mallet-Bachelier, 1859, see p. 48.
%H A060295 D. H. Lehmer, <a href="https://doi.org/10.1090/S0025-5718-43-99091-X">Table to many places of decimals</a>, Queries-Replies, Math. Comp., Vol. 1, No. 1 (1943), pp. 30-31.
%H A060295 Tito Piezas III <a href="http://web.archive.org/web/20230326021805/http://sites.google.com/site/tpiezas/ramanujan">The Ramanujan pages</a>, see section 05.
%H A060295 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/ramanujan.txt">exp(pi*sqrt(163)) to 5000 digits</a>.
%H A060295 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap79.html">exp(Pi*sqrt(163)), the Ramanujan number, to a precision of 2000 digits</a>. [Broken link]
%H A060295 C. Radoux, <a href="http://web.archive.org/web/20150105171125/http://translate.google.com/translate?hl=en&amp;sl=fr&amp;u=http://users.skynet.be/radoux/163.htm">A Formula of Ramanujan (Text in French)</a>.
%H A060295 C. Radoux, <a href="http://web.archive.org/web/20131114015730/http://translate.google.com/translate?hl=en&amp;sl=fr&amp;u=http://users.skynet.be/radoux/163-2.htm">A Formula of Ramanujan (Continued) (Text in French)</a>.
%H A060295 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanConstant.html">Ramanujan Constant</a>.
%H A060295 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A060295 exp(Pi*sqrt(163)) = A199743(6)^3 + 744 - 7.4992... * 10^-13. - _Charles R Greathouse IV_, Jul 15 2020
%e A060295 The Ramanujan number = 262537412640768743.99999999999925007259719818568887935...
%t A060295 RealDigits[N[E^(Pi*Sqrt[163]), 110]][[1]]
%o A060295 (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
%o A060295 (Magma) R:= RealField(); Exp(Pi(R)*Sqrt(163)); // _G. C. Greubel_, Feb 15 2018
%Y A060295 Cf. A058292, A019297, A093436, A102912, A169624, A181045, A181165, A181166.
%K A060295 nonn,easy,cons
%O A060295 18,1
%A A060295 _Jason Earls_, Mar 24 2001