Each of X₁, X₂ and X₃ represents a nonzero digit of the 3-digit base M positive integer X₁X₂X₃; where X₁, X₂ and X₃ are not necessarily distinct.

Determine the respective minimum and maximum positive integer value of M, with M < 100, such that this equation has at least one valid solution.

X₁^X₁ + X₂^X₂ + X₃^X₃ = X₁X₂X₃ - 1

Note: X₁X₂X₃ denotes the concatenation of the three digits.

DEFDBL A-Z

FOR m = 2 TO 100

FOR x1 = 1 TO m - 1
  v1 = x1 * m * m
  x1p = INT(x1 ^ x1 + .5)
FOR x2 = 1 TO m - 1
  v2 = v1 + x2 * m
  x2p = INT(x2 ^ x2 + .5)
FOR x3 = 1 TO m - 1
  v3 = v2 + x3
  x3p = INT(x3 ^ x3 + .5)

  IF x1p + x2p + x3p = v3 - 1 THEN
    PRINT m, x1; x2; x3, v3
  END IF

NEXT
NEXT
NEXT

NEXT

finds only one solution:

4             3  2  3       59
base      representation   decimal value

3^3 + 2^2 + 3^3 = 59 - 1 (decimal representation)

Posted by Charlie on 2009-10-22 18:00:31