Number Theory

2007 Allen Klinger

Number theory topics are easy to understand. A German children's book that presents the essence of this mathematical subject became a best-selling adult title. A charming English translation [1] presents many of the key terms in this area, though in inventive instead of traditional mathematical language. But this follows a long tradition. Fermat, as a judge, and Benjamin Franklin, diplomat and statesman, were taken with the properties of numbers (the latter is said to have been enchanted by magic squares). In more recent times a simply stated problem involving odd and even (tripling and adding one to odd numbers, dividing even ones by two: see below) caused real problems at a computer center.

The following lists some central number theory terms and ideas.

  1. 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.

  2. Although no odd perfect number has ever been found, it is an open problem whether one exists.


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

  4. The fundamental theorem of arithmetic states that any number has a unique factorization into the product of primes.

  5. Diophantine equations involve relationships among unknowns that are integer. (The subject requires searching for a solution in positive integers.)

    Example: Stark [2] gives the following

    x2 -1141y2 = 1

    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



    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
    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.

  6. In 1742 Goldbach conjectured that every even integer greater than 2 is the sum of two primes.

    This has been verified for all even numbers less than 100,000 but no one has been able to prove it [2, p. 5]. Recently Richstein [4] published the verification that this Goldbach Conjecture (the Strong Goldbach Conjecture) holds for n up to
    4 x 1014 .

  7. The Riemann Zeta function is:

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

  8. This function equals the product over all primes p of

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

  9. Ramanujan was without formal mathematical training. However [3] includes this:

    Ramanujan's problem of solutions to


     2N   - 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.


  10. Another Ramanujan problem:

    Find all solutions of n! + 1 = x2.

  11. Davis/Hersh [5] state that "There are nine prime numbers between 9,999,900 and 10,000,000: 9,999,901; 9,999,907; 9,999,929; 9,999,931; 9,999,937; 9,999,943; 9,999,971; 9,999,973; 9,999,991. But among the next hundred integers, from 10,000,000 to 10,000,100, there are only two: 10,000,019 and 10,000,079."
12. Nicomachus theorem


[1] Hans Magnus Enzensberger, 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] Harold M. Stark, An Introduction to Number Theory, Second printing, 1979, MIT Press paperback edition, 1978.


[3] 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.

Read AI Memo 239.

[4] Richstein, J. "Verifying the Goldbach Conjecture up to 4 .1014." Math. Comput. 70, 1745-1750, 2001.

[5] Davis, Philip J. and Hersh, Reuben, The Mathematical Experience, Boston, MA: Birkhauser, 1981.


Singh, Simon, Fermat's Enigma NY: Random House Anchor Books, 1997.

Number Theory Resources Dario Alpern

Tools: On-Line Calculators

More On-Line Calculators

From One to Nine Thousand Ninety-Nine Erich Friedman

Scientific Calculator for Mac PCalc


3n+1 Problem Pythagorean Triples Euclid proved there are infinitely many Pythagorean triples [Singh 1997, p. 293] Primitive Triangles Not Known

8/01/16 Version
©2012 Allen Klinger