# 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): $5777$