Determine the smallest positive integer which is a palindrome in precisely 3 consecutive bases (excluding base 10). What are the next two smallest positive integers with this property?

*** Any solution must have more than one digit in any given base. So, trivial solutions like (0)_{base 3}, (2)_{base 5} or, (C)_{base 14} are not allowed.

*** "Excluding base 10" means that the sequence of bases 9,11,12 for example, does not count as a possibility and, the bases have to be truly consecutive and all be on one side or the other of 10.

(In reply to re: computer solution by Jer)

Ignoring the sporadic:

  300     7    6  0  6
  300     8    4  5  4
  300     9    3  6  3

the rest are all of the form:

ax^2+(a+1)x+a= (a-1)(x+1)^2+(a)(x+1)+(a-1) = (a-2)(x+2)^2+(a+2)x+(a-2)

suggesting the substitution a=(x+2)/2 [1]

And indeed:

(x+2)/2x^2+((x+2)/2+1)x+(x+2)/2 = ((x+2)/2-1)(x+1)^2+((x+2)/2)(x+1)+((x+2)/2-1) = ((x+2)/2-2)(x+2)^2+((x+2)/2+2)(x+2)+((x+2)/2-2) for any even x (x>4).

 

Edited on December 5, 2013, 3:00 am

Posted by broll on 2013-12-05 02:51:14