johndcook.com
Fibonacci product
The product of four consecutive Fibonacci numbers equals the product of two consecutive integers.
For example,
3 × 5 × 8 × 13 = 39 × 40.
I ran across this theorem in a note [1] that says “The product of any four consecutive Fibonacci numbers is twice a triangular number.” Since triangular numbers have the form _n_(_n_ + 1)/2, twice a triangular number is the product of two consecutive integers.
The note also gives a way to find the numbers on the right hand side. We have
_F_ _n_ _F_ _n_ +1 _F_ _n_ +2 _F_ _n_ +3 = _m_(_m_ + 1)
where _m_ equals
_F_ _n_ +1 _F_ _n_ +2
if _n_ is odd and
_F_ _n_ _F_ _n_ +3
if _n_ is even.
In the example at the top, 3 is the 4th Fibonacci number, so _n_ = 4. Since 4 is even, _m_ is the product of the 4th and 7th Fibonacci numbers, i.e. _m_ = 3 × 13 = 39.
## More Fibonacci posts
* Fibonacci meets Pythagoras
* Certified Fibonacci numbers
* Turning trig identities into Fibonacci identities
[1] K. B. Subramaniam. On a link between Triangular and Fibonacci numbers. The Mathematical Gazette, Vol. 103, No. 558 (November 2019), p. 489.
