Prime-Generating Quadratics of form 2 a squared plus p

Theorem

The quadratic form:

$2 a^2 + p$

yields prime numbers for $a = 0, 1, \ldots, p - 1$ for values of $p$:

$3, 5, 11, 29$

Proof

3

\(\ds 2 \times 0^2 + 3\)

\(=\)

\(\ds 0 + 3\)

\(\ds \)

\(=\)

\(\ds 3\)

which is prime

\(\ds 2 \times 1^2 + 3\)

\(=\)

\(\ds 2 + 3\)

\(\ds \)

\(=\)

\(\ds 5\)

which is prime

\(\ds 2 \times 2^2 + 3\)

\(=\)

\(\ds 2 \times 4 + 3\)

\(\ds \)

\(=\)

\(\ds 8 + 3\)

\(\ds \)

\(=\)

\(\ds 11\)

which is prime

5

\(\ds 2 \times 0^2 + 5\)

\(=\)

\(\ds 0 + 5\)

\(\ds \)

\(=\)

\(\ds 5\)

which is prime

\(\ds 2 \times 1^2 + 5\)

\(=\)

\(\ds 2 + 5\)

\(\ds \)

\(=\)

\(\ds 7\)

which is prime

\(\ds 2 \times 2^2 + 5\)

\(=\)

\(\ds 2 \times 4 + 5\)

\(\ds \)

\(=\)

\(\ds 8 + 5\)

\(\ds \)

\(=\)

\(\ds 13\)

which is prime

\(\ds 2 \times 3^2 + 5\)

\(=\)

\(\ds 2 \times 9 + 5\)

\(\ds \)

\(=\)

\(\ds 18 + 5\)

\(\ds \)

\(=\)

\(\ds 23\)

which is prime

\(\ds 2 \times 4^2 + 5\)

\(=\)

\(\ds 2 \times 16 + 5\)

\(\ds \)

\(=\)

\(\ds 32 + 5\)

\(\ds \)

\(=\)

\(\ds 37\)

which is prime

11

\(\ds 2 \times 0^2 + 11\)

\(=\)

\(\ds 0 + 11\)

\(\ds \)

\(=\)

\(\ds 11\)

which is prime

\(\ds 2 \times 1^2 + 11\)

\(=\)

\(\ds 2 + 11\)

\(\ds \)

\(=\)

\(\ds 13\)

which is prime

\(\ds 2 \times 2^2 + 11\)

\(=\)

\(\ds 2 \times 4 + 11\)

\(\ds \)

\(=\)

\(\ds 8 + 11\)

\(\ds \)

\(=\)

\(\ds 19\)

which is prime

\(\ds 2 \times 3^2 + 11\)

\(=\)

\(\ds 2 \times 9 + 11\)

\(\ds \)

\(=\)

\(\ds 18 + 11\)

\(\ds \)

\(=\)

\(\ds 29\)

which is prime

\(\ds 2 \times 4^2 + 11\)

\(=\)

\(\ds 2 \times 16 + 11\)

\(\ds \)

\(=\)

\(\ds 32 + 11\)

\(\ds \)

\(=\)

\(\ds 43\)

which is prime

\(\ds 2 \times 5^2 + 11\)

\(=\)

\(\ds 2 \times 25 + 11\)

\(\ds \)

\(=\)

\(\ds 50 + 11\)

\(\ds \)

\(=\)

\(\ds 61\)

which is prime

\(\ds 2 \times 6^2 + 11\)

\(=\)

\(\ds 2 \times 36 + 11\)

\(\ds \)

\(=\)

\(\ds 72 + 11\)

\(\ds \)

\(=\)

\(\ds 83\)

which is prime

\(\ds 2 \times 7^2 + 11\)

\(=\)

\(\ds 2 \times 49 + 11\)

\(\ds \)

\(=\)

\(\ds 98 + 11\)

\(\ds \)

\(=\)

\(\ds 109\)

which is prime

\(\ds 2 \times 8^2 + 11\)

\(=\)

\(\ds 2 \times 64 + 11\)

\(\ds \)

\(=\)

\(\ds 128 + 11\)

\(\ds \)

\(=\)

\(\ds 139\)

which is prime

\(\ds 2 \times 9^2 + 11\)

\(=\)

\(\ds 2 \times 81 + 11\)

\(\ds \)

\(=\)

\(\ds 162 + 11\)

\(\ds \)

\(=\)

\(\ds 173\)

which is prime

\(\ds 2 \times 10^2 + 11\)

\(=\)

\(\ds 2 \times 100 + 11\)

\(\ds \)

\(=\)

\(\ds 200 + 11\)

\(\ds \)

\(=\)

\(\ds 211\)

which is prime

29

\(\ds 2 \times 0^2 + 29\)

\(=\)

\(\ds 0 + 29\)

\(\ds \)

\(=\)

\(\ds 29\)

which is prime

\(\ds 2 \times 1^2 + 29\)

\(=\)

\(\ds 2 + 29\)

\(\ds \)

\(=\)

\(\ds 31\)

which is prime

\(\ds 2 \times 2^2 + 29\)

\(=\)

\(\ds 2 \times 4 + 29\)

\(\ds \)

\(=\)

\(\ds 8 + 29\)

\(\ds \)

\(=\)

\(\ds 37\)

which is prime

\(\ds 2 \times 3^2 + 29\)

\(=\)

\(\ds 2 \times 9 + 29\)

\(\ds \)

\(=\)

\(\ds 18 + 29\)

\(\ds \)

\(=\)

\(\ds 47\)

which is prime

\(\ds 2 \times 4^2 + 29\)

\(=\)

\(\ds 2 \times 16 + 29\)

\(\ds \)

\(=\)

\(\ds 32 + 29\)

\(\ds \)

\(=\)

\(\ds 61\)

which is prime

\(\ds 2 \times 5^2 + 29\)

\(=\)

\(\ds 2 \times 25 + 29\)

\(\ds \)

\(=\)

\(\ds 50 + 29\)

\(\ds \)

\(=\)

\(\ds 79\)

which is prime

\(\ds 2 \times 6^2 + 29\)

\(=\)

\(\ds 2 \times 36 + 29\)

\(\ds \)

\(=\)

\(\ds 72 + 29\)

\(\ds \)

\(=\)

\(\ds 101\)

which is prime

\(\ds 2 \times 7^2 + 29\)

\(=\)

\(\ds 2 \times 49 + 29\)

\(\ds \)

\(=\)

\(\ds 98 + 29\)

\(\ds \)

\(=\)

\(\ds 127\)

which is prime

\(\ds 2 \times 8^2 + 29\)

\(=\)

\(\ds 2 \times 64 + 29\)

\(\ds \)

\(=\)

\(\ds 128 + 29\)

\(\ds \)

\(=\)

\(\ds 157\)

which is prime

\(\ds 2 \times 9^2 + 29\)

\(=\)

\(\ds 2 \times 81 + 29\)

\(\ds \)

\(=\)

\(\ds 162 + 29\)

\(\ds \)

\(=\)

\(\ds 191\)

which is prime

\(\ds 2 \times 10^2 + 29\)

\(=\)

\(\ds 2 \times 100 + 29\)

\(\ds \)

\(=\)

\(\ds 200 + 29\)

\(\ds \)

\(=\)

\(\ds 229\)

which is prime

\(\ds 2 \times 11^2 + 29\)

\(=\)

\(\ds 2 \times 121 + 29\)

\(\ds \)

\(=\)

\(\ds 242 + 29\)

\(\ds \)

\(=\)

\(\ds 271\)

which is prime

\(\ds 2 \times 12^2 + 29\)

\(=\)

\(\ds 2 \times 144 + 29\)

\(\ds \)

\(=\)

\(\ds 288 + 29\)

\(\ds \)

\(=\)

\(\ds 317\)

which is prime

\(\ds 2 \times 13^2 + 29\)

\(=\)

\(\ds 2 \times 169 + 29\)

\(\ds \)

\(=\)

\(\ds 338 + 29\)

\(\ds \)

\(=\)

\(\ds 367\)

which is prime

\(\ds 2 \times 14^2 + 29\)

\(=\)

\(\ds 2 \times 196 + 29\)

\(\ds \)

\(=\)

\(\ds 392 + 29\)

\(\ds \)

\(=\)

\(\ds 421\)

which is prime

\(\ds 2 \times 15^2 + 29\)

\(=\)

\(\ds 2 \times 225 + 29\)

\(\ds \)

\(=\)

\(\ds 450 + 29\)

\(\ds \)

\(=\)

\(\ds 479\)

which is prime

\(\ds 2 \times 16^2 + 29\)

\(=\)

\(\ds 2 \times 256 + 29\)

\(\ds \)

\(=\)

\(\ds 512 + 29\)

\(\ds \)

\(=\)

\(\ds 541\)

which is prime

\(\ds 2 \times 17^2 + 29\)

\(=\)

\(\ds 2 \times 289 + 29\)

\(\ds \)

\(=\)

\(\ds 578 + 29\)

\(\ds \)

\(=\)

\(\ds 607\)

which is prime

\(\ds 2 \times 18^2 + 29\)

\(=\)

\(\ds 2 \times 324 + 29\)

\(\ds \)

\(=\)

\(\ds 648 + 29\)

\(\ds \)

\(=\)

\(\ds 677\)

which is prime

\(\ds 2 \times 19^2 + 29\)

\(=\)

\(\ds 2 \times 361 + 29\)

\(\ds \)

\(=\)

\(\ds 722 + 29\)

\(\ds \)

\(=\)

\(\ds 751\)

which is prime

\(\ds 2 \times 20^2 + 29\)

\(=\)

\(\ds 2 \times 400 + 29\)

\(\ds \)

\(=\)

\(\ds 800 + 29\)

\(\ds \)

\(=\)

\(\ds 829\)

which is prime

\(\ds 2 \times 21^2 + 29\)

\(=\)

\(\ds 2 \times 441 + 29\)

\(\ds \)

\(=\)

\(\ds 882 + 29\)

\(\ds \)

\(=\)

\(\ds 911\)

which is prime

\(\ds 2 \times 22^2 + 29\)

\(=\)

\(\ds 2 \times 484 + 29\)

\(\ds \)

\(=\)

\(\ds 968 + 29\)

\(\ds \)

\(=\)

\(\ds 997\)

which is prime

\(\ds 2 \times 23^2 + 29\)

\(=\)

\(\ds 2 \times 529 + 29\)

\(\ds \)

\(=\)

\(\ds 1058 + 29\)

\(\ds \)

\(=\)

\(\ds 1087\)

which is prime

\(\ds 2 \times 24^2 + 29\)

\(=\)

\(\ds 2 \times 576 + 29\)

\(\ds \)

\(=\)

\(\ds 1152 + 29\)

\(\ds \)

\(=\)

\(\ds 1181\)

which is prime

\(\ds 2 \times 25^2 + 29\)

\(=\)

\(\ds 2 \times 625 + 29\)

\(\ds \)

\(=\)

\(\ds 1250 + 29\)

\(\ds \)

\(=\)

\(\ds 1279\)

which is prime

\(\ds 2 \times 26^2 + 29\)

\(=\)

\(\ds 2 \times 676 + 29\)

\(\ds \)

\(=\)

\(\ds 1352 + 29\)

\(\ds \)

\(=\)

\(\ds 1381\)

which is prime

\(\ds 2 \times 27^2 + 29\)

\(=\)

\(\ds 2 \times 729 + 29\)

\(\ds \)

\(=\)

\(\ds 1458 + 29\)

\(\ds \)

\(=\)

\(\ds 1487\)

which is prime

\(\ds 2 \times 28^2 + 29\)

\(=\)

\(\ds 2 \times 784 + 29\)

\(\ds \)

\(=\)

\(\ds 1568 + 29\)

\(\ds \)

\(=\)

\(\ds 1597\)

which is prime

When $x = p$ we have:

\(\ds 2 p^2 + p\)

\(=\)

\(\ds p \left({2 p + 1}\right)\)

which has divisors $p$ and $2 p + 1$ and so is not prime.

$\blacksquare$

Sources

• 1989: Paulo Ribenboim: The Book of Prime Number Records (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $29$