Goldbach's Lesser Conjecture

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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

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