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 $1$ modulo $q$ where $q$ itself is a prime can be established by the following elegant modification of Euclid’s proof (http://planetmath.org/ProofThatThereAreInfinitelyManyPrimes).
Let $f(n)=\frac{{n}^{q}-1}{n-1}=1+n+{n}^{2}+\mathrm{\cdots}+{n}^{q-1}$. Let $n>1$ be an integer, and suppose $p\mid f(n)$. Then ${n}^{q}\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$ which implies by Lagrange’s theorem that either $q\mid p-1$ or $n\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$. In other words, every prime divisor^{} of $f(n)$ is congruent to $1$ modulo $q$ unless $n$ is congruent to $1$ modulo that divisor^{}.
Suppose there are only finitely many primes that are congruent to $1$ modulo $q$. Let $P$ be twice their product^{}. Note that $P\equiv 2\phantom{\rule{veryverythickmathspace}{0ex}}(modq)$. Let $p$ be any prime divisor of $f(P)$. If $p\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modq)$, then $p\mid P$ which contradicts $f(P)\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modP)$. Therefore, by the above $P\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$. Therefore $f(P)\equiv 1+P+{P}^{2}+\mathrm{\cdots}+{P}^{q-1}\equiv 1+1+1+\mathrm{\cdots}+1\equiv q\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$. Since $q$ is prime, it follows that $p=q$. Then $P\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$ implies $P\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(modq)$. However, that is inconsistent with our deduction^{} that $P\equiv 2\phantom{\rule{veryverythickmathspace}{0ex}}(modq)$ above. Therefore the original assumption that there are only finitely many primes congruent to $1$ modulo $q$ is false.
References
- 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 |
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Canonical name | SpecialCaseOfDirichletsTheoremOnPrimesInArithmeticProgressions |
Date of creation | 2013-03-22 14:35:38 |
Last modified on | 2013-03-22 14:35:38 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 9 |
Author | bbukh (348) |
Entry type | Theorem |
Classification | msc 11N13 |