Fibonacci product
The Endeavour 2026-09-16
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
Fn Fn+1 Fn+2 Fn+3 = m(m + 1)
where m equals
Fn+1 Fn+2
if n is odd and
Fn Fn+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.
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