Theorem: Prime-Generating Quadratics of form 2 a squared plus p/29

The quadratic form:
:$2 a^2 + 29$
yields prime numbers for $a = 0, 1, \ldots, 28$.

Proof:
{{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 29
      | r = 0 + 29
}}
{{eqn | r = 29
      | c = which is prime
}}
{{eqn | l = 2 \times 1^2 + 29
      | r = 2 + 29
      | c = 
}}
{{eqn | r = 31
      | c = which is prime
}}
{{eqn | l = 2 \times 2^2 + 29
      | r = 2 \times 4 + 29
      | c = 
}}
{{eqn | r = 8 + 29
      | c = 
}}
{{eqn | r = 37
      | c = which is prime
}}
{{eqn | l = 2 \times 3^2 + 29
      | r = 2 \times 9 + 29
      | c = 
}}
{{eqn | r = 18 + 29
      | c = 
}}
{{eqn | r = 47
      | c = which is prime
}}
{{eqn | l = 2 \times 4^2 + 29
      | r = 2 \times 16 + 29
      | c = 
}}
{{eqn | r = 32 + 29
      | c = 
}}
{{eqn | r = 61
      | c = which is prime
}}
{{eqn | l = 2 \times 5^2 + 29
      | r = 2 \times 25 + 29
      | c = 
}}
{{eqn | r = 50 + 29
      | c = 
}}
{{eqn | r = 79
      | c = which is prime
}}
{{eqn | l = 2 \times 6^2 + 29
      | r = 2 \times 36 + 29
      | c = 
}}
{{eqn | r = 72 + 29
      | c = 
}}
{{eqn | r = 101
      | c = which is prime
}}
{{eqn | l = 2 \times 7^2 + 29
      | r = 2 \times 49 + 29
      | c = 
}}
{{eqn | r = 98 + 29
      | c = 
}}
{{eqn | r = 127
      | c = which is prime
}}
{{eqn | l = 2 \times 8^2 + 29
      | r = 2 \times 64 + 29
      | c = 
}}
{{eqn | r = 128 + 29
      | c = 
}}
{{eqn | r = 157
      | c = which is prime
}}
{{eqn | l = 2 \times 9^2 + 29
      | r = 2 \times 81 + 29
      | c = 
}}
{{eqn | r = 162 + 29
      | c = 
}}
{{eqn | r = 191
      | c = which is prime
}}
{{eqn | l = 2 \times 10^2 + 29
      | r = 2 \times 100 + 29
      | c = 
}}
{{eqn | r = 200 + 29
      | c = 
}}
{{eqn | r = 229
      | c = which is prime
}}
{{eqn | l = 2 \times 11^2 + 29
      | r = 2 \times 121 + 29
      | c = 
}}
{{eqn | r = 242 + 29
      | c = 
}}
{{eqn | r = 271
      | c = which is prime
}}
{{eqn | l = 2 \times 12^2 + 29
      | r = 2 \times 144 + 29
      | c = 
}}
{{eqn | r = 288 + 29
      | c = 
}}
{{eqn | r = 317
      | c = which is prime
}}
{{eqn | l = 2 \times 13^2 + 29
      | r = 2 \times 169 + 29
      | c = 
}}
{{eqn | r = 338 + 29
      | c = 
}}
{{eqn | r = 367
      | c = which is prime
}}
{{eqn | l = 2 \times 14^2 + 29
      | r = 2 \times 196 + 29
      | c = 
}}
{{eqn | r = 392 + 29
      | c = 
}}
{{eqn | r = 421
      | c = which is prime
}}
{{eqn | l = 2 \times 15^2 + 29
      | r = 2 \times 225 + 29
      | c = 
}}
{{eqn | r = 450 + 29
      | c = 
}}
{{eqn | r = 479
      | c = which is prime
}}
{{eqn | l = 2 \times 16^2 + 29
      | r = 2 \times 256 + 29
      | c = 
}}
{{eqn | r = 512 + 29
      | c = 
}}
{{eqn | r = 541
      | c = which is prime
}}
{{eqn | l = 2 \times 17^2 + 29
      | r = 2 \times 289 + 29
      | c = 
}}
{{eqn | r = 578 + 29
      | c = 
}}
{{eqn | r = 607
      | c = which is prime
}}
{{eqn | l = 2 \times 18^2 + 29
      | r = 2 \times 324 + 29
      | c = 
}}
{{eqn | r = 648 + 29
      | c = 
}}
{{eqn | r = 677
      | c = which is prime
}}
{{eqn | l = 2 \times 19^2 + 29
      | r = 2 \times 361 + 29
      | c = 
}}
{{eqn | r = 722 + 29
      | c = 
}}
{{eqn | r = 751
      | c = which is prime
}}
{{eqn | l = 2 \times 20^2 + 29
      | r = 2 \times 400 + 29
      | c = 
}}
{{eqn | r = 800 + 29
      | c = 
}}
{{eqn | r = 829
      | c = which is prime
}}
{{eqn | l = 2 \times 21^2 + 29
      | r = 2 \times 441 + 29
      | c = 
}}
{{eqn | r = 882 + 29
      | c = 
}}
{{eqn | r = 911
      | c = which is prime
}}
{{eqn | l = 2 \times 22^2 + 29
      | r = 2 \times 484 + 29
      | c = 
}}
{{eqn | r = 968 + 29
      | c = 
}}
{{eqn | r = 997
      | c = which is prime
}}
{{eqn | l = 2 \times 23^2 + 29
      | r = 2 \times 529 + 29
      | c = 
}}
{{eqn | r = 1058 + 29
      | c = 
}}
{{eqn | r = 1087
      | c = which is prime
}}
{{eqn | l = 2 \times 24^2 + 29
      | r = 2 \times 576 + 29
      | c = 
}}
{{eqn | r = 1152 + 29
      | c = 
}}
{{eqn | r = 1181
      | c = which is prime
}}
{{eqn | l = 2 \times 25^2 + 29
      | r = 2 \times 625 + 29
      | c = 
}}
{{eqn | r = 1250 + 29
      | c = 
}}
{{eqn | r = 1279
      | c = which is prime
}}
{{eqn | l = 2 \times 26^2 + 29
      | r = 2 \times 676 + 29
      | c = 
}}
{{eqn | r = 1352 + 29
      | c = 
}}
{{eqn | r = 1381
      | c = which is prime
}}
{{eqn | l = 2 \times 27^2 + 29
      | r = 2 \times 729 + 29
      | c = 
}}
{{eqn | r = 1458 + 29
      | c = 
}}
{{eqn | r = 1487
      | c = which is prime
}}
{{eqn | l = 2 \times 28^2 + 29
      | r = 2 \times 784 + 29
      | c = 
}}
{{eqn | r = 1568 + 29
      | c = 
}}
{{eqn | r = 1597
      | c = which is prime
}}
{{end-eqn}}