Two well-known results in number theory are:
Fermat's $4n+1$ theorem: Every prime of the form $4n+1$ can be represented as $a^2+b^2 (a,b \in\mathbb{N})$.
Euler's $6n+1$ theorem: Every prime of the form $6n+1$ can be represented as $a^2+3b^2 (a,b \in\mathbb{N})$.
Looking at the Mathworld entries on these theorems here and here, I notice that representation of primes of the form $4n+1$ is stated to be unique (up to order), but that there is no mention of uniqueness in respect of representation of primes of the form $6n+1$. Uniqueness does however seem to hold at least for small primes of this form.
Question: Is the representation of any prime of the form $6n+1$ as $a^2+3b^2$ essentially unique?
If this is the case then a reference to a proof would be appreciated.
from Hot Weekly Questions - Mathematics Stack Exchange
Adam Bailey
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