Goldbach's Lesser Conjecture/Historical Note

Historical Note on Goldbach's Lesser Conjecture

Christian Goldbach conjectured in a letter to Leonhard Paul Euler dated $18$ November $1752$ that all odd integers are expressible in the form $2 a^2 + p$, for $a \ge 0$ and $p$ prime.

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 Moritz Abraham Stern and his students tested all the primes to $9000$, and found the counterexamples $5777$ and $5993$.

Sources

• 1856: Moritz A. Stern: Sur un assertion de Goldbach relative aux nombres impairs (Nouv. Ann. Math. Vol. 15: pp. 23 – 24)

• 1993: Laurent Hodges: A Lesser-Known Goldbach Conjecture (Math. Mag. Vol. 66: pp. 45 – 47)  www.jstor.org/stable/2690477