A060295 - OEIS

%I #89 Sep 08 2024 11:24:38

%S 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 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 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 Decimal expansion of exp(Pi*sqrt(163)).

%C From _Alexander R. Povolotsky_, Jun 23 2009, Apr 04 2012: (Start)

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

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

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

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

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

%C 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 From _Alexander R. Povolotsky_, Oct 16 2010, Apr 04 2012: (Start)

%C 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 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 From _Amiram Eldar_, Jun 24 2021: (Start)

%C This constant was discovered by Hermite (1859).

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

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

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

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

%D Dimitris Vathis, Letter to _N. J. A. Sloane_, Apr 22 1985.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 142.

%H Harry J. Smith, <a href="/A060295/b060295.txt">Table of n, a(n) for n = 18..20000</a>

%H 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 Richard E. Borcherds, <a href="https://www.youtube.com/watch?v=a9k_QmZbwX8">MegaFavNumbers 262537412680768000</a>, video (2020).

%H 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 Alex Clark and Brady Haran, <a href="https://www.youtube.com/watch?v=DRxAVA6gYMM">163 and Ramanujan Constant</a>, Numberphile video (2012).

%H 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 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 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 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 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 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 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 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/ramanujan.txt">exp(pi*sqrt(163)) to 5000 digits</a>.

%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanConstant.html">Ramanujan Constant</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

%e The Ramanujan number = 262537412640768743.99999999999925007259719818568887935...

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

%o (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 (Magma) R:= RealField(); Exp(Pi(R)*Sqrt(163)); // _G. C. Greubel_, Feb 15 2018

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

%K nonn,easy,cons

%O 18,1

%A _Jason Earls_, Mar 24 2001