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.

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