A093372 -id:A093372 - cp's OEIS frontend

A090820 Composite n such that Fibonacci(n) == Legendre(n,5) (mod n).

25, 60, 120, 125, 180, 240, 300, 360, 480, 540, 600, 625, 660, 720, 840, 900, 960, 1080, 1200, 1320, 1440, 1500, 1620, 1680, 1800, 1860, 1920, 1980, 2160, 2400, 2460, 2520, 2640, 2700, 2760, 2880, 3000, 3060, 3125, 3240, 3300, 3360, 3420, 3600, 3660, 3720

Offset: 1

Eric Rowland, Apr 29 2004

nonn

If n is a prime, not 5, then Fibonacci(n) == Legendre(n,5) (mod n) (see for example G. H. Hardy and E. M. Wright, Theory of Numbers).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Masataka Yorinaga, On a congruencial property of Fibonacci numbers (numerical experiments), Math. J. Okayama Univ. 19 (1976/77), no. 1, 5-10.
Masataka Yorinaga, On a congruencial property of Fibonacci numbers (considerations and remarks), Math. J. Okayama Univ. 19 (1976/77), no. 1, 11-17.

Cf. A049062, A093372, A094063.

Select[ Range[ 2, 5000 ], ! PrimeQ[ # ] && Mod[ Fibonacci[ # ] - JacobiSymbol[ #, 5 ], # ] == 0 & ]

N=10^4; for(n=2,N, if(Mod((fibonacci(n)), n)==kronecker(n,5) && !isprime(n), print1(n, ", ")));

More terms from Felix Fröhlich, Apr 24 2014

A094063 Composite n such that Fibonacci(n) == Legendre(n,5) == -1 (mod n).

Original entry on oeis.org

5777, 10877, 12958, 17302, 40948, 41998, 75077, 88198, 95038, 100127, 113573, 130942, 133742, 156178, 160378, 161027, 162133, 163438, 168838, 203942, 219742, 231703, 300847, 314158, 336598, 390598, 393118, 430127, 467038, 480478, 534508

Offset: 1

Eric Rowland, Apr 29 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..235

Cf. A049062, A090820, A093372.

Select[ Range[ 2, 100000 ], ! PrimeQ[ # ] && Mod[ Fibonacci[ # ] - JacobiSymbol[ #, 5 ], # ] == 0 && JacobiSymbol[ #, 5 ] == -1 & ]

More terms from Ryan Propper, Nov 07 2006

A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

Original entry on oeis.org

4181, 6721, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 90061, 96049, 97921, 118441, 146611, 163081, 186961, 197209, 219781, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 330929, 399001, 433621, 438751, 489601, 512461, 520801

Offset: 1

Jianing Song, Sep 12 2018

nonn

Complement of A212423 with respect to A212424.
Intersection of A212424 and A047209.
Composite k == 1,4 (mod 5) such that Fibonacci(k) == 1 (mod k) and that k divides Fibonacci(k-1).

			4181 = 37*113 is composite, while Fibonacci(4180) == 0 (mod 4181), Fibonacci(4181) == 1 (mod 4181), so 4181 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site)
Jon Grantham, Frobenius pseudoprimes, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data.
A. Rotkiewicz, Lucas and Frobenius Pseudoprimes, Annales Mathematicae Silesiane, 17 (2003): 17-39.
Eric Weisstein's World of Mathematics, Frobenius Pseudoprime.

Cf. A047209, A093372, A094394, A094401, A212423, A212424.

for(n=2,500000,if(!isprime(n) && (n%5==1||n%5==4) && fibonacci(n-kronecker(5,n))%n==0 && (fibonacci(n)-kronecker(5,n))%n==0, print1(n, ", ")))

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A090820 Composite n such that Fibonacci(n) == Legendre(n,5) (mod n).

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A094063 Composite n such that Fibonacci(n) == Legendre(n,5) == -1 (mod n).

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A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

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