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 A Perfect Square Problem (Posted on 2006-06-03)
Find all possible pairs of positive integers (x,y) such that both x²+5y and y²+5x are perfect squares.

 See The Solution Submitted by K Sengupta No Rating

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 computer solution--no proof (spoiler) | Comment 4 of 10 |

The program finds these 4 solutions:

x   y      x^2+5y  sqrt   y^2+5x  sqrt
4  4         36  6        36  6
11  27        256  16       784  28
56  69        3481  59     5041  71
77  192       6889  83    37249  193

Of course x and y can be interchanged.

The program:

DEFDBL A-Z
FOR t = 1 TO 1000000
FOR x = 1 TO t / 2
y = t - x
a = x * x + 5 * y
b = y * y + 5 * x
sra = INT(SQR(a) + .5)
IF sra * sra = a THEN
srb = INT(SQR(b) + .5)
IF srb * srb = b THEN
PRINT x; y, a; sra, b; srb
END IF
END IF
NEXT
NEXT

The program was halted at a total for x and y of  about 19,000 without finding other solutions.

 Posted by Charlie on 2006-06-03 16:19:34

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