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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