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
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