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, 143, 1763, 1891, 4181, 5183, 5777, 6601, 6721, 8149, 10403, 10877, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 36863, 40501, 51841, 64079, 64681, 67861, 68101, 68251, 75077, 78409, 79523, 88601, 88831, 90061, 96049, 97343, 97921

Offset

1,2

Comments

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.

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.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Zhi-Hong Sun, Congruences for Fibonacci Numbers [PDF] (Lecture notes, 2009)

Maple

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;

Prog

(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

Crossrefs

Sequence in context: A035304 A241924 A185514 * A159054 A135946 A217534

Adjacent sequences: A220289 A220290 A220291 * A220293 A220294 A220295

Keyword

nonn

Author

Gary Detlefs, Dec 09 2012

Extensions

Missing a(30) added by Charles R Greathouse IV, Dec 25 2012

Name corrected by Jianing Song, Sep 12 2018

Status

approved