Friday, August 25, 2006

Regular Primes: Next Steps

Ernst Kummer presented his proof that Fermat's Last Theorem is true for regular primes in April of 1847. This is a brilliant proof that introduced the idea of ideal numbers and was the biggest advance in Fermat's Last Theorem up to that point.

Still, there were many questions that were raised: how does one determine the class number for a set of cyclotomic integers that are not characterized by unique factorization? How does one determine when Condition (B) applies for a set of cyclotomic integers. Do irregular primes exist?

Kummer announced that Johann Dirichlet had already found a formula for the class number and would show Dirichlet's work could be applied to the class number for cyclotomic integers. Kummer would later prove that Condition (A) implies Condition (B). Interestingly, out of this, would come a connection between class number and Bernoulli numbers (a prime is regular if it does not divide any of the Bernoulli numbers). Irregular primes do exist and it would later be proved that there are an infinite number of them.

All of this work was done between May and September of 1847. Harold M. Edwards has called this a "an extraordinary tour de force." (Harold M. Edwards, Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory) There turn out to be only 3 irregular primes less than 100, they are: 37, 59, and 67 so Kummer's proof was later used to show that Fermat's Last Theorem is true for integers less than 100. Using techniques from this proof, in 1993, Fermat's Last Theorem was proved to be true for all integers less than 4,000,000. (Of course, all these techniques were superceded when Andrew Wiles presented the first version of his famous proof on June 23, 1993.)

To dive into the advances made with regard to regular primes, it is first necessary to review the Euler Product Formula and show how this formula lead's to Dirichlet's formula for class number. I will show that if a prime does not divide its class number, then this shows that a prime is regular (in other words, using the definition of regular primes, Condition (A) implies Condition (B)) and finally, I will show that 37 is an irregular prime.

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