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Omni-MATH / Let p be a prime number. Prove the following theorem of Euler: the equation p=x²+3y² has a solution with x,y∈Z if and only if p=3 or p equiv 1(bmod 3). (You may use the fact that t…

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problem

Let pp be a prime number. Prove the following theorem of Euler: the equation p=x2+3y2p=x^{2}+3 y^{2} has a solution with x,y∈Zx, y \in \mathbb{Z} if and only if p=3p=3 or $p \equiv 1(\bmod 3)$. (You may use the fact that the ring of integers of Q(−3)\mathbb{Q}(\sqrt{-3}) is a principal ideal domain.)
Plain-text mathematical notation (without MathML)
Let p be a prime number. Prove the following theorem of Euler: the equation p=x²+3y² has a solution with x,y∈Z if and only if p=3 or $p \equiv 1(\bmod 3)$. (You may use the fact that the ring of integers of Q(√(−3)) is a principal ideal domain.)
Original LaTeX notation
Let $p$ be a prime number. Prove the following theorem of Euler: the equation $p=x^{2}+3 y^{2}$ has a solution with $x, y \in \mathbb{Z}$ if and only if $p=3$ or $p \equiv 1(\bmod 3)$. (You may use the fact that the ring of integers of $\mathbb{Q}(\sqrt{-3})$ is a principal ideal domain.)

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Source and history

task_id
9b4ae0db-b54c-55a4-9fbb-b3f27d8e38b8
task_key
test--9b4ae0db-b54c-55a4-9fbb-b3f27d8e38b8
task_revision_id
1
config / split
/ test

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