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#3
11-17-2019, 05:39 PM
 Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 86,367
OK, start from the first equation, a^2 - b^2 = ab ± 1. Since a and b are both three-digit numbers, ab is going to be five to seven digits, meaning that that ±1 is almost irrelevant. So to a very good approximation, a^2 - ab + b^2 = 0. Defining x == a/b, this means that x^2 - x - 1 ~= 0, or x ~= (1+sqrt(5))/2 ~= 1.61803 (incidentally, the Golden Ratio, phi). In other words, a ~= 1.61803 * b, or a-b ~= 0.61803*b is a perfect square, or b is a three-digit perfect square times 1.61803 . That only leaves us nine numbers to check, 100 to 324 (any higher square would leave a and/or b with four digits).

Just computing the b values corresponding to those perfect squares, I notice very quickly that phi*144 is very, very close to an integer. So let's start there. phi*144 = 232.99632, so call it 233. And 233*phi = 377.001919, also really close to an integer. So it looks really likely that our numbers will be 233 and 377.

377^2 - 233^2 = 142129 - 54289 = 87840, and 233*377 = 87841. Looks like we've got it.