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A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

%I A244801 #56 Feb 22 2022 00:04:51
%S A244801 1,1,1,3,5,3,16,3,15,26,25,13,39,39,16,28,10,48,7,55,58,49,21,5,37,97,
%T A244801 22,24,26,60,13,64,58,117,120,60,44,160,44,130,174,131,94,31,141,5,
%U A244801 112,3,154,18,29,5,182,250,2,105
%N A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.
%C A244801 A value of 0 indicates a Wall-Sun-Sun prime. No such prime is known and if one exists it is > 4*10^16 (cf. PrimeGrid WSS statistics).
%H A244801 Felix Fröhlich, <a href="/A244801/b244801.txt">Table of n, a(n) for n = 1..291043</a>
%H A244801 F. G. Dorais and D. Klyve, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html">A Wieferich Prime Search up to 6.7 x 10^15</a>, J. Integer Seq. Volume 14, Issue 9 (2011).
%H A244801 R. J. McIntosh and E. L. Roettger, <a href="http://dx.doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Math. Comp. 76 (2007), 2087-2094.
%H A244801 PrimeGrid, <a href="http://prpnet.primegrid.com:13001/">Wall-Sun-Sun Prime Search statistics</a>
%H A244801 PrimeGrid, <a href="http://www.primegrid.com/forum_thread.php?id=3008&amp;nowrap=true#45946">Welcome to the Wall-Sun-Sun prime search</a>
%t A244801 A= 0; p = 0; While[A < 200, p = NextPrime[p];  A= Mod[(Fibonacci[p-JacobiSymbol[p,5]])/p, p]; Print[A]] (* _Javier Rivera Romeu_, Jan 11 2022 *)
%o A244801 (PARI) forprime(p=2, 10^2, a=fibonacci(p-kronecker(p, 5))%p^2; a=a/p; print1(a, ", "))
%o A244801 (Sage)
%o A244801 A, p = 0, 0
%o A244801 while A <200:
%o A244801   p = next_prime(p)
%o A244801   A = (fibonacci(p-legendre_symbol(p, 5))/p)%p
%o A244801   print(A, end=", ") #_Javier Rivera Romeu_, Jan 08 2022
%Y A244801 Cf. A000045, A113650, A195988.
%K A244801 nonn
%O A244801 1,4
%A A244801 _Felix Fröhlich_, Jul 06 2014