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3. Prove that there are no nonzero solutions of a + 6² = 3c, 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 a? + y? = 3 has no rational points. %3D

3. Prove that there are no nonzero solutions of a + 6² = 3c, 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 a? + y? = 3 has no rational points. %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 9AEXP
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3. 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 a? + y? = 3 has no rational points. %3D
Transcribed Image Text:3. 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 a? + y? = 3 has no rational points. %3D
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