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

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justaguide eNotes educator| Certified Educator

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

llltkl | Student

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.

llltkl | Student

The Fibonacci Sequence is the series of numbers:


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.