special case of Dirichlet’s theorem on primes in arithmetic progressions
The special case of Dirichlet’s theorem for primes in arithmetic progressions for primes congruent to modulo where itself is a prime can be established by the following elegant modification of Euclid’s proof (http://planetmath.org/ProofThatThereAreInfinitelyManyPrimes).
Let . Let be an integer, and suppose . Then which implies by Lagrange’s theorem that either or . In other words, every prime divisor of is congruent to modulo unless is congruent to modulo that divisor.
Suppose there are only finitely many primes that are congruent to modulo . Let be twice their product. Note that . Let be any prime divisor of . If , then which contradicts . Therefore, by the above . Therefore . Since is prime, it follows that . Then implies . However, that is inconsistent with our deduction that above. Therefore the original assumption that there are only finitely many primes congruent to modulo is false.
- 1 Henryk Iwaniec and Emmanuel Kowalski. Analytic Number Theory, volume 53 of AMS Colloquium Publications. AMS, 2004.
|Title||special case of Dirichlet’s theorem on primes in arithmetic progressions|
|Date of creation||2013-03-22 14:35:38|
|Last modified on||2013-03-22 14:35:38|
|Last modified by||bbukh (348)|