Determine all possible value(s) of a positive integer x such that the base ten number 14....4 is a perfect square, where the digit 4 is repeated precisely x times.

(In reply to computer exploration--possible solution (spoiler) by Charlie)

I agree with your findings (x = 2 and x = 3), but I believe that some analysis is needed to prove that these are the only solutions. For example, we could prove that if 144… has more than three 4s then it can’t be a square, as follows:

Writing             14444… = (1000a + b)²     for integers a, b, with b < 1000
gives                 14444… = 1000000a² + 2000ab + b²
so that              b² = 444 (mod 2000)

A computer search can quickly show that this equation has no solution for b < 1000, so that the values x = 2 and x = 3, corresponding to sqrt 144 and sqrt 1444 are the only solutions to the problem.


<o:p></o:p>

Posted by Harry on 2009-08-29 22:55:10