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From: ntt@dciem.UUCP (Mark Brader)
Newsgroups: net.math
Subject: Re: prime numbers (high school math)
Message-ID: <442@dciem.UUCP>
Date: Mon, 24-Oct-83 14:51:40 EDT
Article-I.D.: dciem.442
Posted: Mon Oct 24 14:51:40 1983
Date-Received: Mon, 24-Oct-83 15:23:00 EDT
References: <5587@arizona.UUCP>
Organization: NTT Systems Inc., Toronto, Canada
Lines: 21

Tim Budd (arizona!budd) writes, in part:
    
           ... more difficult, however, and I have never seen a proof
    of it (although I will admit I have never looked!), is that prime
    numbers greater than 5 (along with a lot of other numbers) can be
    expressed as sqrt(1 + 24n).  Try it!  By the way, the sequence n which
    when plugged into here results in prime numbers is itself interesting,
    but has no more pattern to it than prime numbers themselves.

First, 5 itself is also of this form, with n=1.

I couldn't remember seeing this before, but proved it easily.
As you know, modulo 6, all prime numbers greater than or equal to 5
are equal (congruent, if you prefer) to 1 or 5.  Therefore, modulo 24,
they must be equal to (case 1) 1, 7, 13, or 19, or (case 5) 5, 11, 17, or 23.
But modulo 24, 23 = -1, 19 = -5, etc.  So we can write these choices as
+-1, +-5, +-7, +-11 (where +- is a plus-or-minus sign).  The squares of
these are 1, 25, 49, 121, all of which modulo 24 are equal to 1.  This
is equivalent to the result above.

Mark Brader, NTT Systems Inc., Toronto; decvax!utzoo!dciem!ntt