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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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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