From: William C Waterhouse <William.C.Waterhouse@mizar.ursa.fi>
Newsgroups: sci.math
Subject: Re: 163 magic (Re: A series problem involving the number 41)
Date: 31 Oct 1997 21:26:10 GMT

From: wcw@math.psu.edu (William C Waterhouse)

In article <01bce16d$c2d11b20$cc6246c3@sun2.alpin.or.at>,
"Martin Mooslechner" <mooslechner@salzburg.co.at> writes:
>...
> Is there an 'easy' way to explain how the fact that the Euler pn produces
> so many primes is related to the largest
> associated-quadratic-number-ring-with-unique-factorization-number 163 ?

Here's the argument in one direction (from memory, so it may not be
in the most polished form).

1) Let t = (-1 + Sqrt(-163)) / 2.  Then the elements in the number
ring have the form a+bt with a and b in Z.

2) The norm of a+bt is (a-b/2)^2 + (163/4)b^2. Thus the norm of any
element with b nonzero is at least 41.

3) In particular, the norm of x + t is  x^2 - x + 41. For 0 < x < 41,
this number is less than (41)^2. Hence if it is not prime, it is
divisible by a prime  p  less than 41.

4) Now consider the principal ideal I generated by x + t.  It factors
into prime ideals, each of which is either given by a rational prime
or a complex prime ideal with prime norm.  Clearly no rational
prime divides x+t, so one of the factors of I must be a prime ideal
with norm p.

5) But by 2), there is no element with norm  p,  and so the prime
ideal cannot be principal.

6) Thus, if we know unique factorization, the norms of all these
x+t must be prime.