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): $5777$