Subject: Quadratic reciprocity for the millions
In article <3m1928$ib2@senator-bedfellow.MIT.EDU>,
tycchow@ATHENA.MIT.EDU (Timothy Y Chow) writes:
|> Last night I found myself trying to explain why quadratic reciprocity
|> is such a great theorem to some engineering students who knew nothing
|> about number theory. It was quite a challenge. Finally I came up with
|> the following idea (based on an example that I saw in a class I took
|> from Goro Shimura): if one runs the following one-line Maple program---
|>
|> for n from 3 to 100 do ifactor(n^2-5) od;
|>
|> (which prints out the prime factorization of all numbers of the form
|> n^2 - 5 as n ranges from 3 to 100)---then one observes that one never
|> gets any primes ending in 3 or 7. This, of course, is easily explained
|> using quadratic reciprocity.
[Explicitly: if p is an odd prime dividing n^2 - 5 then the Legendre symbol
(5|p) = 1 and by q.r. 1 = (5|p) = (p|5), so p = 0, 1, or 4 mod 5, so p
can't end in 3 or 7. Of course 2 doesn't end in 3 or 7 either.]