No Primes of Form a^2+3b^2 and 3k-1

No prime of the form  
\[a^2+3b^2\]
  can also be written in the form  
\[3k+2\]
.
To see why, working modulo 3, a square number must be congruent to either 0 or 1, so
\[a^2+3b^2 \equiv 0+3 \times 0 \; (mod \; 3) \equiv 0 \; (mod \; 3)\]

\[a^2+3b^2 \equiv 1+3 \times 0 \; (mod \; 3) \equiv 1 \; (mod \; 3)\]

\[a^2+3b^2 \equiv 0+3 \times 1 \; (mod \; 3) \equiv 0 \; (mod \; 3)\]

\[a^2+3b^2 \equiv 1+3 \times 1 \; (mod \; 3) \equiv 1 \; (mod \; 3)\]

But  
\[3k+2 \equiv 2 \; (mod \; 3)\]
  so no such prime exists.

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