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A Perfect Square Problem (Posted on 2006-06-03) Difficulty: 4 of 5
Find all possible pairs of positive integers (x,y) such that both x+5y and y+5x are perfect squares.

See The Solution Submitted by K Sengupta    
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Solution 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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