Consecutive Numbers in the Fibbonacci Sequence and Sums of Two Squares

[Imaginer1]

I've noticed lots of interesting properties of the patterns of numbers in the Fibbonacci sequence that can be expressed as the sum of two squares. In fact, it's what got me into number theory in the first place. There seem to be no two adjacent entries that are not the sum of two squares- and it seems that no sum of two squares is surrounded by two entries that are not.

So, formally:

1) There exists no entry in the Fibbonacci sequence F(n) such that neither F(n) or F(n+1) are the sum of two squares

2) There an infinite number of n that F(n), F(n-1) and F(n+1) are all the sum of two squares.

Less formally:

Why do numbers that are the sum of two squares 'like' to group together in twos and threes?

Why do numbers that are not the sum of two squares like to be apart from each other?

 

[Joseph Fermat]

Could you show examples for these conjectures. The way you wrote them is rather difficult to understand. They seem to be very interesting.

 

[dodo]

Hi, Imaginer,
if you see this page (scroll down to equation 58), every other Fibonacci number is the sum of the squares of two previous Fibonacci numbers (for example, 5=1²+2², 13=2²+3², 34=3²+5², ...) (or, if you prefer, the sum of the squares of two consecutive Fibonacci numbers is another Fibonacci number). (The latter statement follows from the more known eq.55 in that webpage, which in turn is not hard to prove by induction on one of the subindices.)

So, it only takes some chance to have one of the other intervening Fibonacci numbers to be also a sum of squares (and there are a lot of them).