Prime-Generating Quadratics of form 2 a squared plus p/11

Theorem

The quadratic form:

$2 a^2 + 11$

yields prime numbers for $a = 0, 1, \ldots, 10$.

Proof

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

\(=\)

\(\ds 0 + 11\)

\(\ds \)

\(=\)

\(\ds 11\)

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

$\blacksquare$