03. Prove that there are no nonzero solutions of a +62 3c2, where a, b, and...

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03. Prove that there are no nonzero solutions of a +62 3c2, where a, b, and c 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 c are even, and this solution is not primitive. Remark: This proof implies that the circle 22+y23 has no rational points. 03. Prove that there are no nonzero solutions of a +62 3c2, where a, b, and c 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 c are even, and this solution is not primitive. Remark: This proof implies that the circle 22+y23 has no rational points.