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A220292 Nonprime n not divisible by 2 or 3 such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n. 1

%I #21 Sep 12 2018 05:18:33

%S 1,143,1763,1891,4181,5183,5777,6601,6721,8149,10403,10877,13201,

%T 13981,15251,17119,17711,30889,34561,36863,40501,51841,64079,64681,

%U 67861,68101,68251,75077,78409,79523,88601,88831,90061,96049,97343,97921

%N Nonprime n not divisible by 2 or 3 such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n.

%C A Fibonacci based primality criterion of Legendre and Lagrange which is listed as theorem 2.2 in the Zhi-Hong Sun link at A000032 states the Fibonacci(p-1) mod p = (1 - Legendre(p/5))/2. This sequence lists the pseudoprimes to this criterion which are not divisible by 2 or 3.

%C The number of pseudoprimes appears to decrease as n increases, there being 36 between 1 and 100,000, 17 between 100,000 and 200,000,and 11 between 200,000 and 300,000.

%H Charles R Greathouse IV, <a href="/A220292/b220292.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Hong Sun, <a href="http://www.hytc.edu.cn/xsjl/szh/ConFn.pdf">Congruences for Fibonacci Numbers</a> [PDF] (Lecture notes, 2009)

%p with(numtheory): with(combinat): for n from 1 to 40000 do if n mod 2 <> 0 and n mod 3 <>0 and fibonacci(n-1) mod n = (1-legendre(n,5))/2 and not isprime(n) then print(n) fi od;

%o (PARI) is(n)=gcd(n,6)==1 && ((Mod([1,1;1,0],n))^(n-1))[1,2]==(1-kronecker(n,5))/2 && !isprime(n) \\ _Charles R Greathouse IV_, Dec 22 2012

%K nonn

%O 1,2

%A _Gary Detlefs_, Dec 09 2012

%E Missing a(30) added by _Charles R Greathouse IV_, Dec 25 2012

%E Name corrected by _Jianing Song_, Sep 12 2018

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)