Triangular Fibonacci Numbers

Theorem

The only Fibonacci numbers which are also triangular are:

$0, 1, 3, 21, 55$

This sequence is A039595 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

$\ds 0$

$=$

$\ds \dfrac {0 \times 1} 2$

$\ds 1$

$=$

$\ds \dfrac {1 \times 2} 2$

$\ds 3$

$=$

$\ds \dfrac {2 \times 3} 2$

$\ds = 1 + 2$

$\ds 21$

$=$

$\ds \dfrac {6 \times 7} 2$

$\ds = 8 + 13$

$\ds 55$

$=$

$\ds \dfrac {10 \times 11} 2$

$\ds = 21 + 34$

It remains to be shown that these are the only ones.

Let $F_n$ be the $n$th Fibonacci number.

From Odd Square is Eight Triangles Plus One, $F_n$ is triangular if and only if $8 F_n + 1$ is square.

It remains to be demonstrated that $8 F_n + 1$ is square if and only if:

$n \in \set{\pm 1, 0, 2, 4, 8, 10}$

So, let $8 F_n + 1$ be square.

Then:

$n \equiv \begin{cases} \pm 1 \pmod {2^5 \times 5} & : n \text { odd} \\ 0, 2, 4, 8, 10 \pmod {2^5 \times 5^2 \times 11} & : n \text { even} \end{cases}$

This theorem requires a proof.
In particular: The proof needs to be studied.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $55$

1989: Luo Ming: On Triangular Fibonacci Numbers (The Fibonacci Quarterly Vol. 27, no. 2: pp. 98 – 108)

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $55$