Admin: Created page with "{{FMM |title=Square Free Sequence |problem= Prove that no number in the sequence 11, 111, 1111, 11111, ... is the square of an integer. |solution= If s is a number in the..."

2019-11-16T22:31:53Z

Created page with "{{FMM |title=Square Free Sequence |problem= Prove that no number in the sequence 11, 111, 1111, 11111, ... is the square of an integer. |solution= If s is a number in the..."

New page

{{FMM
|title=Square Free Sequence
|problem=

Prove that no number in the sequence

11, 111, 1111, 11111, ...

is the square of an integer.

|solution=

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

<math>
11 + 100m
</math>

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:

<math>
11 + 100m = 100m + 8 + 3 = 4 (25m + 2) + 3
</math>

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

<math>
4n^2
</math>

or

<math>
4n^2 + 4n + 1
</math>

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.

}}

FMM10 - Revision history

Admin: Created page with "{{FMM |title=Square Free Sequence |problem= Prove that no number in the sequence 11, 111, 1111, 11111, ... is the square of an integer. |solution= If s is a number in the..."