**8. Repeat A Question - ***Power of the Internet*

A number growing in size (placing one grain on the first square
of a chessboard, doubling the grains on each sucessive square) is a
concept that over centuries formed some key mathematical ideas
(*infinity*, infinite series). Today the Internet fosters rapid
communication. People can jointly discuss an issue using the world's computers.
This has been going on since a visit to UCLA by UCSD Prof. R. Graham May
21, 1999. Here is a general view of that discussion.

There was an open problem when this was initiated on June 24, 1999. It was to find n, * a second integer greater than one*,
with *the property* (8.1) that when 2 is raised to it and 3 is taken away from
the result, the answer is evenly divisible by the n value. What *was known
* is that the value 4700063497 does this. In other words
2^{4700063497}-3 is an integer times 4700063497.

Electronic mail (email) communication put this problem to this page's
author. Inquiry by email led a correspondent suggesting listing the
problem. Posting it to an email list on number theory led to
at least four people sending informative messages in hours. *This note
continues the fact-seeking process * through the Internet. (Informative email
continues to arrive.) A recent message included:
*"D.H. Lehmer found such an n. Look at ***Mathematics of
Computation*** about 10 years ago. ..."* Since then more has been
done.

Computer-communications stimulate learning about individuals involved in such
work. [Related items are in mathematical books.]

Any n satisfying the property (8.1) must be odd.
(Since the minus three causes the expression to be
odd. No even number can exactly divide an odd.) That a solution can't be
a multiple of three (substitute n=3k, find that for the property to be
satisfied 3 has to evenly divide a power of 2 ... which is impossible), nor in several other families of integers is also
known. UCLA Prof. E. Koutsoupias pointed out July 1, 1999 that
Fermat's Little Theorem eliminates the possibility that such an n would be a prime.

This property written
concisely in mathematical notation is:

Experts: Peter Montgomery wrote
* This is problem F10 in the 1981 Edition of Richard K. Guy's
***Unsolved Problems in Number Theory**.
He gives the solution 4700063497 = 19 * 47 * 5263229.

To refer to
it later we write:

4700063497 = 19 * 47 * 5263229 (8.2)
June 11, 1999 Joe Crump pointed out that Number Sequences is an available resource and (8.2) appears in List [list begins a(1) = 1].

On June 24, 1999 Dr. Montgomery conveyed this number to me by electronic mail:

63130707451134435989380140059866138830623361447484274774099906755 (8.3)

On June 29, 1999 he stated that this solves the problem.
He used a computer program Prof. Noam Elkies mentioned on
June 5, 1999. Using sources detailed in Computations,
items others sent me by email, I was able to factor the above number to:

5 * 97 * 130166407115741105132742556824466265630151260716462422214638983 (8.4)

This used *Factoris*, a tool that factors integers and polynomials
written by XIAO Gang (email: xiao@unice.fr). His program concluded that
130166407115741105132742556824466265630151260716462422214638983
is a prime. It is striking that both the numbers found factor into three
primes with two of them small
(except see below).
Further it is also striking
that the
ratio of the two is close to the ratio of their corresponding large
prime factors. (8.3) is about 6.3 * 10^{64}; (8.2) is about 4.7 * 10^{9}. The ratio (8.3)/(8.2) is about 1.34 * 10^{55}. The
ratio of their largest factors is about 2.47 * 10^{55}.

In Chapter 3 of G. H. Hardy's *Ramanujan, Twelve Lectures on
Subjects Suggested By His Life and Work,* Cambridge UK: The University
Press, 1940, a *round number* is said to be one that "is the product
of a considerable number of comparatively small factors." On the second
page of that chapter appears "We find, if we try numbers at random from
near the end of the factor tables, that f(n) [the number of distinct
prime factors of a number n] is usually not 7 or 8 but 3 or 4; ...".

On 11/13/06 Max Alekseyev pointed out that
Two lists
the value 8365386194032363 found by J. Crump in 2000. Alekseyev found
yet
another value, 3468371109448915. {He suggested use of Mathematica's
PowerMod; the online documentation indicates that
PowerMod[a, b, n] gives a^{b} mod n.}

Now the questions about (8.3) begin. [On p. 275
of Guy, Richard K. and Woodrow, Robert E., Eds., *The Lighter Side of
Mathematics*, Mathematical Association of America, Wash. DC, 1994, in
Guy's article "The Strong Law of Small Numbers," there is this item:
D. H. & Emma Lehmer discovered that 2^n = 3 (mod n) for n = 4700063497,
but for no smaller n > 1 .]

*7 December 2006 Version*