A217853 - OEIS

%I #20 May 25 2019 02:02:13

%S 91,7381,597871,48427561,3922632451,317733228541,25736391511831,

%T 2084647712458321,168856464709124011,1107867264956562636991,

%U 588766087155780604365200461,47690053059618228953581237351,25344449488056571213320166359119221,166284933091139163730593611482181209801

%N Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

%C These numbers were obtained for values of k from 1 to 20, with the following exceptions: k = 10, 12, 13, 16, 17, 19, for which were obtained 3^n mod n = 3^7, 3^31, 3^37, 3^25, 3^31, 3^13.

%C Conjecture: There are infinitely many Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8, where k is a natural number.

%C It is true: for example, when 2k+1 is a prime number (see A210461). - _Bruno Berselli_, Jan 22 2013

%H Charles R Greathouse IV, <a href="/A217853/b217853.txt">Table of n, a(n) for n = 1..190</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>

%t Select[Table[(3^(4k + 2) - 1)/8, {k, 80}], PowerMod[3, # - 1, #] == 1 &] (* _Alonso del Arte_, May 14 2019 *)

%o (PARI) list(lim)=my(v=List(),t); lim\=1; for(k=1,(logint(8*lim+1,3)-2)\4, t=3^(4*k + 2)>>3; if(Mod(3,t)^t==3, listput(v,t))); Vec(v) \\ _Charles R Greathouse IV_, Jun 30 2017

%Y Cf. A005935, A210461 (subsequence), A217841.

%K nonn

%O 1,1

%A _Marius Coman_, Oct 12 2012