A219114 Integers n such that n^2 is the difference of two Fibonacci numbers.

0, 1, 2, 4, 9, 12, 15, 24

Offset: 1

Author

Max Alekseyev, Nov 12 2012

Keywords

• nonn
• hard
• more

Comments

Numbers n such that n^2 is in A007298.

No other terms below 10^10000. - Manfred Scheucher, Jun 12 2015

Examples

			The only known square differences of Fibonacci numbers are:
0^2 = F(2)-F(1) = F(k)-F(k) for any k,
1^2 = F(1)-F(0) = F(2)-F(0) = F(3)-F(1) = F(3)-F(2) = F(4)-F(3),
2^2 = F(5)-F(1) = F(5)-F(2),
4^2 = F(8)-F(5),
9^2 = F(11)-F(6),
12^2 = F(12)-F(0) = F(13)-F(11) = F(14)-F(13),
15^2 = F(13)-F(6),
24^2 = F(15)-F(9).
		

Links

• MathOverflow, Can the difference of two distinct Fibonacci numbers be a square infinitely often?
• Manfred Scheucher, Sage Script

Crossrefs

Cf. A000045 (Fibonacci numbers).

Cf. A007298 (differences of Fibonacci numbers).

Programs

Mathematica

t = Union[Flatten[Table[Fibonacci[n] - Fibonacci[i], {n, 100}, {i, n}]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2] (* T. D. Noe, Feb 12 2013 *)