Find all integer solution of the equation $3a^2-b^2=1$. It is my problem.I seemed easy but I have no way to do this.
Find $a,b$ in $3a^2-b^2=1$
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algebra-precalculus
number-theory
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4Is the right-hand side 0 (as in the title) or 1 (as in the body of the question)? – 2017-01-12
2 Answers
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Note that $$3a^2-b^2 =1 \implies b^2 +1=3a^2 \implies b^2 \equiv -1 \pmod 3$$ This is a contradiction, as no square can be $-1$ modulo $3$, as shown through $$(3k)^2 \equiv 0 \pmod 3$$$$(3k \pm 1)^2 \equiv 3(3k^2 \pm 2k )+ 1 \equiv 1 \pmod 3$$
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since $$3a^2=b^2+1$$ use well know $$b^2+1\equiv1, 2\pmod 4$$ and $$3a^2\equiv 0,3 \pmod 4$$