Examples
An integer is a prime for the Gaussian integers if it is a prime number (in the normal sense) that is congruent to 3 modulo 4. The primes of the type 4n + 3 are
- 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ….
They correspond to the following values of n:
- 0, 1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 20, 25, 26, 31, 32, 34, 37, 40, 41, 44, 47, 49, 52, 55, 56, 59, 62, 65, 67, 70, 76, 77, 82, 86, 89, 91, 94, 95, ….
The strong form of Dirichlet's theorem implies that
is a divergent series.
The following table lists several arithmetic progressions and the first few prime numbers in each of them.
| Arithmetic progression |
First 10 of infinitely many primes | OEIS sequence |
|---|---|---|
| 2n + 1 | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … | A065091 |
| 4n + 1 | 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, … | A002144 |
| 4n + 3 | 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, … | A002145 |
| 6n + 1 | 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, … | A002476 |
| 6n + 5 | 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, … | A007528 |
| 8n + 1 | 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, … | A007519 |
| 8n + 3 | 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, … | A007520 |
| 8n + 5 | 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, … | A007521 |
| 8n + 7 | 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, … | A007522 |
| 10n + 1 | 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, … | A030430 |
| 10n + 3 | 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, … | A030431 |
| 10n + 7 | 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, … | A030432 |
| 10n + 9 | 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, … | A030433 |
Read more about this topic: Dirichlet's Theorem On Arithmetic Progressions
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