budd@arizona.UUCP (tim budd) (10/22/83)
The recent comments on prime numbers recalls to mind a couple facts I stumbled upon concerning prime numbers way back when I was in high school (more years ago than I like to remember). The first is truly trivial to prove, although at the time I was impressed with myself for seeing it, and is that prime numbers can all be expressed (along with a lot of non- prime numbers) as 6n + or - 1. The second "fact" is more difficult, however, and I have never seen a proof of it (although I will admit I have never looked!). That is that prime numbers greater than 5 (again 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.
ntt@dciem.UUCP (Mark Brader) (10/24/83)
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