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 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.The…

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