Goldbach's Lesser Conjecture

False Conjecture

Every positive odd integer $n$ can be expressed in the form:

$n = 2 a^2 + p$

where:

$a \in \Z_{\ge 0}$ is a non-negative integer

$p$ is a prime number or $1$.

Refutation

There are two known counterexamples:

$5777$ and $5993$

as follows:

$5777$ is a Stern Number

The number $5777$ cannot be represented in the form:

$5777 = 2 a^2 + p$

where:

$a \in \Z_{\ge 0}$ is a non-negative integer

$p$ is a prime number.

$5993$ is a Stern Number

The number $5993$ cannot be represented in the form:

$5993 = 2 a^2 + p$

where:

$a \in \Z_{\ge 0}$ is a non-negative integer

$p$ is a prime number.

Source of Name

This entry was named for Christian Goldbach.

Historical Note

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

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5777$
• 1990: C. Ashbacher: Representing Integers as the Sum of a Prime and Twice a Square (J. Recr. Math. Vol. 22: pp. 244 – 245)
• 1993: Laurent Hodges: A Lesser-Known Goldbach Conjecture (Math. Mag. Vol. 66: pp. 45 – 47)  www.jstor.org/stable/2690477
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5777$