Wikipedia:Reference desk/Archives/Mathematics/2020 February 14

= February 14 =

Lowest Fibonacci power of 10?
Not counting 1. What is the lowest power of 10 that is a Fibonacci number?Naraht (talk) 16:15, 14 February 2020 (UTC)
 * There aren't any. Fibonacci number says:
 * The only nontrivial square Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers..
 * PrimeHunter (talk) 16:22, 14 February 2020 (UTC)


 * If I'm right, any Fibonacci number that is divisible by 10 is divisible by 610. Georgia guy (talk) 20:31, 14 February 2020 (UTC)
 * You are right. Using the indexing scheme where F0 = 0, F1 = 1, as in our article on Fibonacci numbers, the following divisibility property holds:
 * If m ≥ 3, m｜n  ⇔  Fm｜Fn.
 * So 10｜Fn ⇔ (2｜Fn) ∧ (5｜Fn) ⇔ (F3｜Fn)  ∧ (F5｜Fn) ⇔ (3｜n)  ∧ (5｜n) ⇔ 15｜n ⇔ 610｜Fn .  --Lambiam 06:43, 15 February 2020 (UTC)