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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)[reply]

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)[reply]

References

^ Cohn, JHE (1964), "Square Fibonacci Numbers etc", Fibonacci Quarterly, 2: 109–13
^ Pethő, Attila (2001), "Diophantine properties of linear recursive sequences II", Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 17: 81–96
^ 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)[reply]

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 ⇔ F_m｜F_n .

So 10｜F_n ⇔ (2｜F_n) ∧ (5｜F_n) ⇔ (F₃｜F_n) ∧ (F₅｜F_n) ⇔ (3｜n) ∧ (5｜n) ⇔ 15｜n ⇔ 610｜F_n . --Lambiam 06:43, 15 February 2020 (UTC)[reply]

[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

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