# it is thought that there are only five Fibonacci numbers that are also triangular numbers. Find them.

### 3 Answers | Add Yours

Does this list end with 5 numbers, is there a proof that it couldn't continue indefinitely.

### Hide Replies ▲

According to this link, there are only four (I don't think repeating 1 should count toward this number). I have no time to think about why though-either that 1989 reference was simply the most convenient or well-known, or it was the first appearance of the proof, in which case it might be a pretty hard proof. ``

In 1989, L. Ming proved the only Fibonacci numbers which are triangular are 1, 3, 21, and 55. Please refer to Ming, L. "On Triangular Fibonacci Numbers." *Fib. Quart.* **27**, 98-108, 1989.

The Fibonacci Sequence is the series of numbers:

0,1,1,2,3,5,8,13,21,34,55,89..............

The next number is found by adding up the two numbers before it.

The general formula is `T_n=T_(n-1)+T_(n-2)`

On the other hand, the Triangular Number Sequence is:

1,3,6,10,15,21,28,36,45,55,.......where `T_n=(n(n+1))/2.`

Five Fibonacci numbers that are also triangular numbers are **1,1,3,21,55.**

**Sources:**