Prove that there are no nonzero solutions of a? + b? = 3c2, where a, b, and e are integers. Hint: It suffices to show that there are no primitive triples (a, b, c) satisfying this equation. Work modulo 4 to show that if there is a solution, then a, b, and e are even, and this solution is not primitive. Remark: This proof implies that the circle 1 + y? = 3 has no rational points.

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