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Least number k such that the decimal representation of 1/k has period Fibonacci(n).
1

%I #21 Jul 09 2023 10:37:23

%S 3,11,27,41,73,53,43,103,1321,497867,323,467,11311,20141,12169,

%T 232159532264041847249

%N Least number k such that the decimal representation of 1/k has period Fibonacci(n).

%C The period of 1/k is the least integer p such that 10^p = 1 (mod k). The integer p is also known as the multiplicative order of 10 (mod k).

%D Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.

%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

%D S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>

%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=FibonacciNumber"> Fibonacci Numbers</a>

%e p(k) is the period of 1/k, we obtain with k=3,11,27,41,73,53,43,103 p(3)=1,p(11)=2,p(27)=3,p(41)=5,p(73)=8, p(53)=13,p(43)=21, p(103)=34

%p For the great numbers (p > 70), the maple program is very slow. That's what we use an process of two steps: factoring 10^p-1 with elliptic curve method (see the first web site), and then, for each factor q(k), k=1,2,...,r computation the periods of 1/q(k) and keep the period q(i) such that q(i) = Fibonacci number. The 17th term required 3h 2m for the computing of (10^1597) -1 T:=array(0..100);U:=array(0..100); n0:=0:n1:=1:T[1] = 1:for i from 2 to 30 do: n2:=n0+n1:T[i]:=n2:n0:=n1:n1:=n2:od:U[1]:=3:U[2]:=3:for q from 3 to 10 do: p0:=T[q]: indic:=0:for n from 1 to 2000 do:for p from 1 to 150 while(irem(10^p,n)<>1 or gcd(n,10)<>1 ) do:od: if irem(10^p,n) = 1 and gcd(n,10) = 1 and p=p0 and indic=0 then U[q]:=n:indic:=1:else fi:od: od: for n from 1 to 10 do:print( U[n]):od:

%Y Cf. A000045, A039834 (signed Fibonacci numbers), A002329, A072859 (periodic sequences), A003060

%K nonn,base

%O 2,1

%A _Michel Lagneau_, Feb 19 2010

%E Edited by _T. D. Noe_, Apr 14 2010