Number Theory

©Allen Klinger

This subject is so playful that reading a
children's book [1] presents much of it, aside from the changed terms.
The author of that work, a philosopher, and his translator, both created
readable works for people of almost all ages. Here is one thing
that both [1] and the subject of this article concern.

For a *perfect number *the
factors (other than the number itself) sum to that value.

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*Example*: 6 factors into 1, 2, 3. The sum
as well as the product of these three values is 6.

A
number that has as factors only itself and one is called *prime*.

The *fundamental
theorem of arithmetic*
[2] states that any number has a unique factorization into the product of primes.

The
Riemann Zeta function [3] is:

S k^{-s }where the sum is from k=1 to infinity and Real(s) >
1.

This function
equals the product over all primes p of

(1-p^{-s})^{-1}
, where again Real(s) > 1.

The
topic of Diophantine equations concerns whether there is a solution in positive
integers for some situation. Stark [2] gives the following example:

x^{2
}-1141y^{2} = 1

He says: We might ask, does (this) have any
solution in positive integers (beginning with the observation that É x =1, y =
0 satisfies)? We see É that

x = sqrt(1141y^{2} + 1)

Thus
the question is: Is 1141 y + 1 ever a perfect square? This may be checked experimentally. It turns out that the answer is no for
all positive y less than 1 million É perhaps we should experiment further. The answer is still no for all y less
than 1 trillion (1 million million, or 10^{12}). We go overboard and check all y up to 1
trillion trillion (10^{24}).
Again the answer is, no. No
one in his right mind would really believe that there could be a positive y
such that x = sqrt(1141y^{2}
+ 1) is an integer if there is no such y less than 1 trillion trillion. But
there is. In fact there are infinitely many of them, the smallest among them
having 26 digits. (Concludes quotation from [2].)

Number
theory topics are easy to understand. Benjamin Franklin was enchanted by magic
squares. Ramanujan was without formal mathematical training. However [4]
includes this:

Ramanujan's
problem of solutions to

2^{N } - 7 = X^{ N}

was
searched to about N = 10^{40}; only his solutions (N = 3, 4, 5, 7, 15)
were found. It has recently been proven that these are the only ones.

Another
Ramanujan problem:

Find
all solutions of n! + 1 = x (Concludes quote from [4].)^{2}.

Ramanujan is responsible for the approximation to
p,
(2143/22)^{1/4}. His life is described in [5].

The idea of residue is similar to modulo, something we use daily when writing
the distinct twenty-four hours of the day with reference to only twelve
numbers. We will write modulo *mod* in expressions like this restatement of* 1400 hours military time is 2 p.m.*:

14 = 2 mod 12

In [2] there are extensive materials on solving algebraic
equations in a different manner than common, using the notion of answers
that are the same modulo some number. That idea is taken to an
extreme in the situation where a simple statement is probed for the
numbers that satisfy it. Here's one such situation.

1. For what values of n integer is it true that nth power of 2 is 3 mod
n?

Mathematics favors terse symbols. By replacing the sign of ordinary equality,
=, with == (two equal signs next to each other), we gain a way of
expressing *equivalent modulo number n* (we just insert == and
append *(mod n)*. Then 1. can become:

2^{n} == 3 (mod n). But this can be written another way also.

2. For what integer values of n is the nth power of 2 minus 3, all
divided by n, also integer? In other words, when does this hold?

[2^{n} - 3]/n is integer.

For answers and insight into work in this area click New or Mod. Additional
material of interest is at Size and Words.

The field consists of many *easily-stated conjectures*. The difficulties in proving these conjectures are
legendary. The game like aspects are present in [6-8] leading toward
*recreational* mathematics. One unsolved problem with a
computational flavor follows.

Lagarias [9] reports everything known up to that point about an
extremely simple to state intractible item he
calls "the 3x+1 problem and all its aliases." It is also known as the Collatz problem. Vardi [10] wrote "The Collatz map is taken to be x -> x/2 if x is even and ... x -> 3x+1 if x is odd." In words from [9]:

The conjecture is that the sequence n, f(n), f(f(n)), ... is ultimately
periodic for all n and such that there is only one final cycle 1 -> 4 -> 2 -> 1.

References

[1] Enzensberger, Hans Magnus, *The Number Devil: A Mathematical Adventure*
(Translated by Michael Henry Heim, Illustrated by Rotraut Susanne
Berner) NY: Metropolitan Books, Henry Holt and Company, 1998.

[2]
Stark, Harold M., *An Introduction to Number Theory*, Second printing, 1979, MIT Press paperback
edition, 1978.

[3] Abramowitz, Milton and Stegun, Irene A. (eds.), *Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables*, National
Bureau of Standards, June 1964, tenth printing December 1972 with
corrections, Superintendent of Documents, U.S. Government Printing
Office, Washington, D.C. 20402.

[4] Beeler,
M., Gosper, R.W., and Schroeppel, R., *Hakmem*, MIT AI Memo 239, Feb. 29, 1972.
Retyped and converted to html
Web browser format by Henry Baker, April,
1995.

[5] Kanigel, Robert, *The Man Who Knew Infinity - A Life of the Genius Ramanujan*,
NY: Simon and Schuster, first published Charles Scribner's Sons, 1991.

[6]
Berlekamp, Elwyn, Conway, John H., and Guy, Richard K., *Winning Ways for
Your Mathematical Plays*,

[7]
Conway, John H., and Guy, Richard K., *The Book of Numbers
*, Springer-Verlag, 1996.

[8]
Guy, Richard K., *Unsolved Problems in Number Theory*
, Springer-Verlag, 1994.

[9]
Lagarias, Jeff, "The 3x+1 problem and its generalizations," *American
Mathematical Monthly*** 92**, (1), 1985, 3-23.

[10]
Vardi, Ilan, *Computational Recreations in Mathematica*, Addison-Wesley
Pub Co., ISBN: 0201529890, April 1991.

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5/13/02 Version |
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©5/10/2002 Allen Klinger |