From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: a+b.sqrt(-d)
Date: Thu, 13 Apr 2000 12:23:55 GMT
Newsgroups: sci.math
Summary: [missing]

In article <955624011.5177.0.nnrp-09.9e989aee@news.demon.co.uk>,
  "Dan Goodman" <dog AT fcbob DOT demon DOT co DOT uk> wrote:
> How can one prove that the numbers of the form a+b.sqrt(-2) where a,b
> integers, is a unique factorisation domain?

That's quite easy. Show that there is a Euclidean algorithm for
this ring of algebraic integers. If
{u, v in R = {a + b sqrt(-2), a, b in Z}
and v is nonzero, then there exists w in R with |u - wv| < |v|.
To find w, consider u/v. If we "round" its real and imaginary parts
to get something in R; that something is w.

> More generally, how can you work
> out that d=163 is the largest prime where a+b.sqrt(-d) is a UFD?

Actually {a + b sqrt{-163}: a, b in Z}
isn't a UFD, but
{(a + b sqrt{-163})/2: a, b in Z, a - b even}
is a UFD. Proving this is hard is a naive way, but easy with some of
the basic methods of algebraic number theory; see any text on this
topic. Proving that 163 is the last prime that works is very difficult.
See for instance David Cox's book, _Primes of the form x^2 + ny^2_.

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


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==============================================================================

From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: getting easy primes
Date: 29 May 2000 23:00:27 GMT
Newsgroups: sci.math
Summary: [missing]


|:41, 43, 47, 53, 61, 71, 83, 97 ........
...
>n^2+n+41 has been known about for ages.

In Harvey Cohn's book Advanced Number Theory, he has a section titled
"The Famous Polynomials x^2+x+q". Letting f_q(x)=x^2+x+q, he proves
the theorem:

   The polynomial f_q(x) will assume prime values for 0<=x<=q-2 if
   and only if for d=1-4q, R(sqrt(d)) has class number 1.

Note that d is the discriminant of the polynomial, R(sqrt(d)) is the
field of elements of the form r+s*sqrt(d) where r and s are rational,
and that "class number 1" is equivalent to unique factorization for
the ring of integers, which are in the case of d=1 (mod 4) the elements
of the form m+n*((1+sqrt(d))/2) where m and n are integers. The
negative integer d with largest magnitude such that R(sqrt(d)) has
class number 1 is d=-163 which corresponds to x^2+x+41.

Keith Ramsay
==============================================================================

From: landsbur@troi.cc.rochester.edu (Steven E. Landsburg)
Subject: Re: almost relationships between pi and e
Date: 31 Dec 2000 04:06:58 GMT
Newsgroups: sci.math
Summary: [missing]

Here's a sketch of why e^(pi*sqrt(163)) is so close to an integer.
This all has to do with unique factorization and elliptic curves,
so I'm sure James Harris will be particularly interested. 

Let w be a complex number and let E be the elliptic curve C/(Z + Zw).
Write q = exp(2 pi i w).  Let j(E) be the j-invariant of E.  

Fact I:  j(E) = 1/q + 744 + (a power series in q with integer coefficients
                             that get big pretty fast)

Corollary:  When |q| is small, j(E) is close to 1/q + 744.

Fact II:  Let K be a quadratic extension of Q, and let (1,w) be a Z-basis
for the ring of integers in K.  Then j(w) is an algebraic integer of degree
equal to the class number of K.

Corollary:  If the ring of integers in K is a UFD, then j(w) is a rational
integer.

Fact III:  The ring of integers in Q(sqrt(-163)) is a UFD.

Now let w = (1+sqrt(-163))/2 and argue as follows:

   1)  By Facts II and III, j(w) is a rational integer.
   2)  By Fact I, j(w) is close to 1/q + 744, with 
                       
               q = - exp(- pi sqrt(163))
   
   3)  By 1) and 2), 1/q = exp(pi sqrt(163)) is close to a rational
       integer.

To make this argument complete, one needs proofs of Facts I, II and
III.  Fact I is from the Fourier expansion of the function j(w) that
assigns to w the j-invariant of the elliptic curve j(Z+Zw).  Fact II
comes from the theory of complex multiplication.  Fact III is much 
easier and is standard in algebraic number theory textbooks.    
        

Steven E. Landsburg
http://www.courses.rochester.edu/landsburg/ 
==============================================================================

From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: almost relationships between pi and e
Date: 01 Jan 2001 01:15:28 GMT
Newsgroups: sci.math

In article <978200235.28154.0.nnrp-09.9e989aee@news.demon.co.uk>,
"Dan Goodman" <dog AT fcbob DOT demon DOT co DOT uk> writes:
|(pi^4+pi^5)^(1/6)=e
|
|Well, almost (try it). Apparently, e^(pi * sqrt(163)) is an integer
|(almost). Is there any deep reason for these two results, or is just
|coincidence?

It looks like the first one is mere coincidence, but the second one
is for a beautful reason. The imaginary quadratic field Q(sqrt(-163))
has a ring of integers which is a unique factorization domain (class
number 1). As a result of some interesting number theory, the value
of the "j" function at a related point is an integer. The main term
of the series for j at that point is e^{pi*sqrt(163)}. The next term
is 744 as I recall. And then there's the "tail", whose sum is small
(terms in powers of e^{-pi*sqrt(163)}).

Keith Ramsay