From johnbcos@iol.ie Fri Jan 8 18:01:22 1999
Date: Thu, 7 Jan 1999 10:56:33 -0500
From: John B. Cosgrave <johnbcos@iol.ie>
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: a 'millennium' prime
I would like to announce the discovery (quite fortuitously) of - what I
would like to call - a millennium prime, namely a prime with exactly 2000
digits (wait for it! It comes later, below).
This is how it happened.
I teach a third year course B.Ed./B.A. undergraduate course on Number
Theory and Cryptography (a number of my Maple worksheets from that
course have been put up by David Joyner of the US Naval Academy at
this Web address: http://web.usna.navy.mil/~wdj/crypto.htm - I don't have
a Web site at my own College - the Administration ... ).
In particular, I teach a number of primality tests to my students. In the
first three years (1995-98) of that course I taught them - with complete
proof - what I like to call the 'Lucas-Kraitchik-Lehmer-Selfridge'
theorem, and over the past few days I have been putting together some
Maple work to present in the coming term on Pocklington's theorem (a
version of it). [This will be first time to attempt to teach this theorem
to my students. I also intend teaching Proth's theorem for the first
time.] I have put together a Maple worksheet called '1914_pap.mws', in
which I have gone back to Pocklington's original 1914-16 paper, and worked
through it with numerical illustrations. [I also believe that I have made
his presentation more understandable, at least to myself!!]
A version of one of Pocklington's theorems which I will present to my
students is this:
Let N - 1 = p^alpha*U, where p is prime, p^alpha is the largest power of p
dividing (N - 1), and p^alpha > U. Suppose that:
1. a^(N -1) = 1 (mod N) for some integer a, and that
2. gcd(a^((N - 1)/p) - 1, N) = 1,
then N is prime.
In a separate Maple worksheet - pockling.mws - I wanted to put together
some numerical illustrations (quirky and beautiful examples - not humdrum
ones) of the above theorem, and - from a whole range of possible numbers -
I decided to subject the following numbers (f[n]) to Pocklington's
theorem:
f[1] = 2*3 + 1,
f[2] = 2*3*5^2 + 1,
f[3] = 2*3*5*7^3 + 1,
in general: f[n] = p[1]*p[2]* ... *p[n]*p[n+1]^n + 1, (p[i], the
i-th.prime)
where the 'p' of Pocklington's theorem is p[n+1], and the 'U' is
p[1]*p[2]* ... *p[n].
I began my search (fixing 'a' at 2) on Jan. 5th. (my 53rd. birthday), and
by the end of that day I had found the following examples:
f[n] is proved prime by Pocklington's theorem for the following values of
n (testing the range 1 to 180): 1, 2, 10, 20 and 173.
The largest prime of those, p[173], has 952 digits.
Of course I wanted to press on, in the hope that I would be able to
present this year's 3rd. years with a titanic prime.
[Last year I was able to present my 3rd. year students (right at the start
of their course, and it created a certain excitement) with two titanic
primes - proved prime by the Lucas-Kraitchik-Lehmer-Selfridge theorem.
Those primes were:
2^50*(p[1]*p[2]*...*p[20])^3 + 1 (having 1006 digits), and
2^37*(1!*2!*3!*...*50!) + 1 (having 1405 digits)]
I pressed on overnight, testing the range 300 to 330, and this morning I
woke to find n=325 produced a prime, which happened to have 2000 digits,
and it was about two hours later before my brain suddenly said: hey! the
millennium! I think it took me so long to make the connection, because,
I have to confess, I'm one of those people who doesn't recognise the next
millennium as starting on Jan 1st. 2000, but rather the following year.
However it will probably interest my students!
Of course I really want to press on and find a 'gigantic' prime but I lack
the computational power, and also I need my computer for my regular work.
You will know (if not, then see Chris Caldwell's remarkable Web site) that
Yves Gallot has written a programme for finding primes using Proth's
theorem. I would dearly love to organise a search for the above type of
prime, and perhaps I might when I finally get a Web site at my College
......
In the meantime, a Happy New Year to you all.
Best wishes,
John
###############################
#
# John B. Cosgrave
# Mathematics Department,
# St. Patrick's College,
# Drumcondra,
# Dublin 9,
# IRELAND.
#
# [St. Patrick's College is a
# College of Dublin City University]
#
# Home e-mail: johnbcos@iol.ie
# College e-mail: John.Cosgrave@spd.ie
#
# Is 2^p - 1 prime for infinitely many p? (I hope so.)
# Is 2^(2^n) + 1 prime for some n > 4? (Surely at least one?)
# Is Pi^e transcendental? (Probably.)
# Is 2^sqrt(2) + 3^sqrt(3) irrational? (Almost certainly.)
# Is zeta(3) a rational multiple of Pi^3? (Hardly.)
#
###############################
###############################
#
# John B. Cosgrave
# Mathematics Department,
# St. Patrick's College,
# Drumcondra,
# Dublin 9,
# IRELAND.
#
# [St. Patrick's College is a
# College of Dublin City University]
#
# Home e-mail: johnbcos@iol.ie
# College e-mail: John.Cosgrave@spd.ie
#
# Is 2^p - 1 prime for infinitely many p? (I hope so.)
# Is 2^(2^n) + 1 prime for some n > 4? (Surely at least one?)
# Is Pi^e transcendental? (Probably.)
# Is 2^sqrt(2) + 3^sqrt(3) irrational? (Almost certainly.)
# Is zeta(3) a rational multiple of Pi^3? (Hardly.)
#
###############################