Difference between revisions of "2004 JBMO Problems/Problem 3"
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Similarly, we get y is congruent to 0 mod 7 | Similarly, we get y is congruent to 0 mod 7 | ||
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Latest revision as of 19:09, 10 April 2019
Let 4x + 3y = a^2 3x + 4y = b^2
Then 7(x + y) = a^2 + b^2
But any perfect square can only be congruent to 0,1,2, or 4 modulo 7
Thus
a = 7p
b = 7q
x + y = 7(p^2 + q^2)
x + y is congruent to 0 mod 7.
4x + 3y = a^2 = 49p^2
3(x+y) + x = 49p^2
x is congruent to 0 mod 7
Similarly, we get y is congruent to 0 mod 7
-igetit
