My Math Forum Diophantine Eq, problem. Help please

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 October 27th, 2010, 09:07 AM #1 Newbie   Joined: Aug 2010 Posts: 17 Thanks: 0 Diophantine Eq, problem. Help please Show that the Equation below has no solution in non-zero rational numbers x,y,z; x^2+y^2= 7z^2. I am interested in the method of proving, the logical steps to the solution.
 October 27th, 2010, 11:45 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Diophantine Eq, problem. Help please Work mod 4.
 October 28th, 2010, 08:48 AM #3 Newbie   Joined: Aug 2010 Posts: 17 Thanks: 0 Re: Diophantine Eq, problem. Help please Prove by contradiction worked fine for me... Let n be the least common denominator of x, y, z, so that a, b, c and n are integers, then (a/n)^2+(b/n)^2 = 7(c/n)^2...by multiplying n^2, we have a^2+b^2 = 7c^2. If a, b, and c have common factor m, then we can replace them by a/m, b/m, and c/m. Suppose a, b, and c have no common factor, then one can reduce the last equation by modulo 7 Call A and B the reductions of Modulo 7. The right side is a multiple of 7 and so it becomes 0, and we are left with A+B = 0 and is satisfied only by the trivial zero, A = 0 and B = 0. But saying A = 0 and B = 0 is the same as saying that A and B are both multiples of 7 and A^2 and B^2 are multiples of 49 and their sum 7c^2 is a multiple of 49 and Thus c^2 is multiple of 7 which in turn indicates that c is multiple of 7. And contrary to our assumption, a, b and c share a common factor..>voilą There is no solution consisting of non-zero rational numbers. Reduction of Module 4, let me try that and see if it works. But is the above reasoning correct and elegant?
 October 28th, 2010, 03:55 PM #4 Global Moderator   Joined: Dec 2006 Posts: 19,963 Thanks: 1849 What if A + B = 7?
 October 28th, 2010, 11:20 PM #5 Newbie   Joined: Aug 2010 Posts: 17 Thanks: 0 Re: Diophantine Eq, problem. Help please Call A and B the reductions of Modulo 7. Another variables just like a and b.
 October 29th, 2010, 02:30 AM #6 Global Moderator   Joined: Dec 2006 Posts: 19,963 Thanks: 1849 A + B = 7 doesn't imply that A = B = 0. You need to show that A and B cannot be 1 and 6 or 2 and 5 or 3 and 4.

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