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.
Example: 6 factors into 1, 2, 3. The sum
as well as the product of these three values is 6.
Example: Stark [2] gives the following
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
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
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.
S k^{-s }where the sum is from k=1 to infinity and Real(s) >
1.
(1-p^{-s})^{-1}
, where again Real(s) > 1.
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.
Find
all solutions of n! + 1 = x^{2}.
[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. http://www.inwap.com/pdp10/hbaker/hakmem/number.html
[4]
Richstein, J. "Verifying the Goldbach Conjecture up to 4
^{.}10^{14}." Math. Comput. 70, 1745-1750, 2001.
[5] Davis, Philip J. and Hersh, Reuben, The Mathematical
Experience, Boston, MA: Birkhauser, 1981.
Notes |
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 | http://www.cs.ucla.edu/~klinger/numbers.htm | |||
©2012 Allen Klinger |