## You are here

Homespecial case of Dirichlet's theorem on primes in arithmetic progressions

## Primary tabs

# 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.

Let $f(n)=\frac{n^{q}-1}{n-1}=1+n+n^{2}+\cdots+n^{{q-1}}$. Let $n>1$ be an integer, and suppose $p\mid f(n)$. Then $n^{q}\equiv 1\;\;(\mathop{{\rm mod}}p)$ which implies by Lagrange’s theorem that either $q\mid p-1$ or $n\equiv 1\;\;(\mathop{{\rm mod}}p)$. 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 $P\equiv 2\;\;(\mathop{{\rm mod}}q)$. Let $p$ be any prime divisor of $f(P)$. If $p\equiv 1\;\;(\mathop{{\rm mod}}q)$, then $p\mid P$ which contradicts $f(P)\equiv 1\;\;(\mathop{{\rm mod}}P)$. Therefore, by the above $P\equiv 1\;\;(\mathop{{\rm mod}}p)$. Therefore $f(P)\equiv 1+P+P^{2}+\cdots+P^{{q-1}}\equiv 1+1+1+\cdots+1\equiv q\;\;(\mathop% {{\rm mod}}p)$. Since $q$ is prime, it follows that $p=q$. Then $P\equiv 1\;\;(\mathop{{\rm mod}}p)$ implies $P\equiv 1\;\;(\mathop{{\rm mod}}q)$. However, that is inconsistent with our deduction that $P\equiv 2\;\;(\mathop{{\rm mod}}q)$ 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.

## Mathematics Subject Classification

11N13*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections