Recurring perfect squares

Number Theory Level 3

How many integers in the infinite sequence 11 , 111 , 1111 , … , 11111 … 1 ⏟ n number of 1’s , … 11, \, 111, \, 1111, \, \ldots, \, \underbrace{11111\ldots1}_{n \text{ number of 1's}} \, , \ldots are perfect square ?


The answer is 0.

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1 solution

Lucas Nascimento
Feb 9, 2017

Perfect squares only leave remainder 1 or 0 when divided by 4, but 111...1 always leaves remainder 3 when divided by 4 because it ends in 11. Soon there is numbers perfect squares in this sequence.

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