Find all possible pairs of positive integers (x,y) such that both x²+5y and y²+5x are perfect squares.
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.
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Posted by Charlie
on 2006-06-03 16:19:34 |
computer solution--no proof (spoiler)
