From charlesreid1

Friday Morning Math Problem

Square Free Sequence

Prove that no number in the sequence

11, 111, 1111, 11111, ...

is the square of an integer.

If s is a number in the sequence, s must have the form

where m is a non-negative integer. This can be rearranged into two parts: the part divisible by 4, and the part not divisible by 4:

which means that when divided by 4, all numbers in the sequence have a remainder of 3.

Furthermore, we know that all squares are of the form


and therefore leave remainders of 0 or 1 when divided by 4.

Thus, a number s in the sequence that always has a remainder of 3 cannot be a square.