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Old 11-18-2019, 10:42 PM
Manlob is offline
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Join Date: May 2000
Posts: 191
Another way to solve this without knowing the Cassini Identity: Let c2=a-b. Eliminate a from the two equations and solve for b, giving
b=(c2ħsqrt(5c4ħ4)/2
For b to be integer, 5c4ħ4 must be a perfect square. Values of c meeting this can be found using the solution to Pell's equation, but it easier to just make a spreadsheet. Put "10" (representing c) in Column 1, sqrt(5c4-4) in Column 2, and sqrt(5c4+4) in Column 3. Drag the three cells down until Column 1 contains 31 (to cover all cases with c having 3 digits). It is immediately clear which value in Column 2 or 3 is the only integer: c=12, and the radical is 322.

Plug into the expression for b and add c2=144 to get a. The ħ results in two solutions:
a=55 and b=-89 (which is rejected because they are not 3 digits)
a=377 and b=233.