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.

 

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:

 

x² -1141y² = 1

 

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

 

    x  = sqrt(1141y² + 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 … perhaps we should experiment further.  The answer is still no for all y less than 1 trillion (1 million million, or 10¹²).  We go overboard and check all y up to 1 trillion trillion (10²⁴).  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(1141y² + 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 = 10⁴⁰; 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:

2ⁿ == 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?

[2ⁿ - 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.