Newsgroups: sci.math From: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock) Subject: Re: "e**(pi*sqrt(163))" Date: Wed, 17 Jun 1992 02:42:34 GMT In article <9206162124.AA18000@ucbvax.Berkeley.EDU> BANDA@AUSVM1.VNET.IBM.COM writes: > Ramanujan once casually claimed that "e**(pi*sqrt(163)) is very close to > an integer". Apparently the reason the above is true has got to do > with quadratic number fields and their class numbers. Can somebody > explain the whole argument......Thanks very much > > Cheers, Venu If t=(-1+sqrt(-163))/2 and x=exp(2*Pi*I*t), then -(64*3*5*23*29)^3=j(x)=1/x+744+196884x+21493760x^2+864299970x^3+... Hence, e^(pi*sqrt(163)) is approximately 744+(64*3*5*23*29)^3. The j here is the j-invariant of the elliptic curve (complex torus) C^2/(Z+Zt). More generally if t is a nonreal quadratic algebraic integer and the ring Z[t] has class number h, then j(x) is an algebraic integer of degree h. Of the 13 complex quadratic extensions of Z with class number 1 Z[(-1+sqrt(-163))/2] has the largest discriminant. See Serre's article on Complex Multiplication in Cassels and Frohlich, Algebraic Number Theory, for the other 12 and some proofs. I think anyone who spends time reading about this subject will be awed by its beauty. -- Bradley W. Brock, Department of Mathematics Rose-Hulman Institute of Technology | "Honor one another above yourselves." brock@nextwork.rose-hulman.edu | -Paul of Tarsus to the Romans ============================================================================== From: hb3@aixterm7.urz.uni-heidelberg.de (Franz Lemmermeyer) Newsgroups: sci.math.symbolic Subject: Re: Exp[Pi*Sqrt[163]] Date: 24 Apr 1995 19:38:46 GMT In article <3ngm38\$44r@controversy.math.lsa.umich.edu>, jackgold@news-server.engin.umich.edu (Jack Goldberg) writes: |> |> Keywords: |> |> Can anyone refer me to a history of the orgin of |> |> Exp[Pi*Sqrt[163]] |> |> a number which appears in many Computer Algebra |> texts as an illustration of the dangers of |> generalizing from scant information (my |> interpretation)? This number is uncannily |> close to a (very large) integer and I have often |> wondered how it was discovered and whether there |> are other such interesting examples. This comes from number theory and is connected with the fact that the imaginary quadratic number field with discriminant -163 has class number 1. There is no discriminant beyond -163 with class number 1, so this is sort of a record. Anyway, if you're looking for material on this subject, try D. Cox's book "Primes of the form x^2-ny^2 ...", or any other source on "Complex Multiplication" (that's what this subject is called) you can find. BTW, two smaller examples are -67 and -43. franz `&' ******************************************************** # Franz Lemmermeyer ** Die endgueltige * # Erwin-Rohde-Str. 19 ** Teilung * _#_ 69120 Heidelberg ** Deutschlands, * ( # ) ** das ist unser * / O \ hb3@ix.urz.uni-heidelberg.de ** Auftrag! * ( === ) ***** Chlodwig Poth **** `---' ********************************************************