Smallest Odd Number not of form 2 a squared plus p/Historical Note

Historical Note on Smallest Odd Number not of form 2 a squared plus p
The question of odd integers expressible in the form $2 a^2 + p$, for $a \ge 0$ and $p$ prime, was initially raised by in a letter to  $18$ November dated $1752$, in which he conjectured that all odd integers were so expressible.

At that time, $1$ was considered to be prime. Thus $1 = 2 \times 0^2 + 1$ and $3 = 2 \times 1^2 + 1$ were considered to fit the criteria, as was $17 = 0^2 + 17$.

The conjecture was believed to hold until $1856$, when and his students tested all the primes to $9000$, and found the counterexamples $5777$ and $5993$.

He then went on to investigate odd numbers, and more specifically prime numbers, that cannot be represented in the form $2 a^2 + p$ where $a > 0$. Seeming to forget about $3$, he stated that the smallest such prime number was $17$.