Determine all possible value(s) of a positive integer N, such that N and N⁴ together contain precisely nine digits from 0 to 9 which are all different. Neither N nor N⁴ can contain any leading zero.

Note: Try to solve this problem analytically, although computer program/spreadsheet solutions are welcome.

The number of digits of N and N⁴ is given as nine, therefore N must be two digits in length. This limit is confirmed as the ceiling of the 4th root of the smallest base-10 seven-digit number, 1000000, is 32 and the floor of the 4th root of the largest base-10 seven-digit number, 9999999, is 56, N⁴, therefore, must be between 32⁴ and 56⁴.

32⁴ = 1048576. With each digit being distinct, this is a solution. After an examination of each of the 4th powers of the numbers between 33 and 56, it is found as the sole solution in base-10.

(45₁₂)⁴ = 2786301₁₂ might also be considered a solution. :-)

Edited on November 2, 2008, 12:59 am

Posted by Dej Mar on 2008-11-01 20:53:35