Rescinded Admission

jimstandard
Wed, 20 Jun 2007 14:18:42 +0000

I will just quote wikipedia since it is a pretty hard thing to define:

In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat’s little theorem, Euler’s theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler’s φ function, integer sequences, factorials and Fibonacci numbers all also fall into this area.

Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include:

The Goldbach conjecture concerning the expression of even numbers as sums of two primes.
Catalan’s conjecture (now Mihăilescu’s theorem) regarding successive integer powers.
The twin prime conjecture about the infinitude of prime pairs.
The Collatz conjecture concerning a simple iteration.
Fermat’s last theorem (stated in 1637, but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that x^n + y^n = z^n for some integer n greater than 2.
The theory of Diophantine equations has even been shown to be undecidable (see Hilbert’s tenth problem).