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A220292
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Nonprime n not divisible by 2 or 3 such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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MAPLE
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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;
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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