Fibonacci

Squares in the Fibonacci series

F₁ = F₂ = 1 and F₁₂ = 144. We ask ourselves whether these are the only squares in the Fibonacci series, and how to prove that.

A positive integer n is always either odd or even, so n = 2k or n = 2k-1 for certain integer k. Then n² = 4k² or n² = 4k²-4k+1. Thus, if we divide n² by 4, the remainder will be either 0 or 1. Apparently a square can never have a remainder 2 or 3 when it is divided by 4.
Now let's divide the Fibonacci numbers by 4. The remainders are:
1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, ...
The numbers 1, 1, 2, 3, 1, 0 are repeated again and again. (note)
We conclude from this argument that no Fibonacci number of the form F_6k-3 or of the form F_6k-2 is a square because when a square is divided by 4, the remainder is 2 or 3.
Each positive integer n may be written in one of the forms 3k, 3k-1 or 3k-2. Then the square n² has the form 9k² or 9k²-6k+1 or 9k²-12k+4. So, if a square is divided by 3, the remainder is unequal to 2 or 3. Divide the Fibonacci numbers by 3. This yields the following remainders:
1 1 2 0 2 2 1 0 1 1 2 0 2 2 1 0 1 1 2 0 2 2 1 0 1 1 ...
The recurring part is 1 1 2 0 2 2 1 0. Thus, the Fibonacci numbers of the form F_8k-5 or F_8k-3 or F_8k-2 are no squares.
We continue this process. Each positive integer n can be written in the form 4k, 4k-1, 4k-2 or 4k-3. The remainder of the square n², if divided by 8 is 0, 1 or 4.
The remainders of the Fibonacci numbers after division by 8 are:
1 1 2 3 5 0 5 5 2 7 1 0 1 1 2 3 5 0 5 5 2 7 1 0 1 1 2 3 5 0 5 5 2 7 1 0 1 1 ...
The recurring part is 1 1 2 3 5 0 5 5 2 7 1 0. If we compare these results we notice that only the Fibonacci numbers of the form F_12k-11, F_12k-10, F_12k-6, F_12k-1 and F_12k may be squares.
This last result is promising, but pursuing this idea does not give the desired progress. Still, the last result can slightly be improved, if we write n as a multiple of 8 plus remainder. Divide the square of n by 16. Then the remainder is 0, 1, 4 or 9. The corresponding Fibonacci remainders are:
1 1 2 3 5 8 13 5 2 7 9 0 9 9 2 11 13 8 5 13 2 15 1 0 1 1 ...
with recurring part 1 1 2 3 5 8 13 5 2 7 9 0 9 9 2 11 13 8 5 13 2 15 1 0. Thus the only Fibonacci numbers that may be squares have the index 24k-23, 24k-22, 24k-13, 24k-12, 24k-11, 24k-10, 24k-1 or 24k. These numbers may be taken together, yielding: 12k-11, 12k-10, 12k-1 or 12k.