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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#Fibonacci primes says:
The only nontrivial square Fibonacci number is 144.[1] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[2] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[3].
PrimeHunter (talk) 16:22, 14 February 2020 (UTC)

References

  1. ^ Cohn, JHE (1964), "Square Fibonacci Numbers etc", Fibonacci Quarterly, 2: 109–13
  2. ^ Pethő, Attila (2001), "Diophantine properties of linear recursive sequences II", Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 17: 81–96
  3. ^ Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969
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)
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