

A307767


The "nonresidue" pseudoprimes: odd composite numbers n such that b(n)^((n1)/2) == 1 (mod n), where base b(n) = A020649(n).


3



3277, 3281, 29341, 49141, 80581, 88357, 104653, 121463, 196093, 314821, 320167, 458989, 476971, 489997, 491209, 721801, 800605, 838861, 873181, 877099, 973241, 1004653, 1251949, 1268551, 1302451, 1325843, 1373653, 1397419, 1441091, 1507963, 1509709, 1530787, 1590751, 1678541, 1809697
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

As is well known, for an odd prime p, b(p) is the smallest quadratic nonresidue b modulo p if and only if b(p) is the smallest base b such that b^((p1)/2) == 1 (mod p). Note that b(n) is always a prime.
Conjecture: If 2^((n1)/2) == 1 (mod n), then b(n) = 2, where b(n) as above. This is true for odd primes n; is it for odd composites n? If so, then all composite numbers n such that 2^((n1)/2) == 1 (mod n) are in this sequence.
It seems that, for defined pseudoprimes n (similar to the odd primes p),
b(n) is the smallest base b such that b^((n1)/2) == 1 (mod n), although this is not required by their definition.
Note: a "nonresidue" pseudoprime n is a strong pseudoprime to base b(n); the Jacobi symbol (b(n)/n) = 1, where b(n) is the smallest nonresidue modulo n; such a pseudoprime n is not a Proth number, so n = k*2^m + 1 with odd k > 2^m.
Problem: are there infinitely many such numbers?


LINKS

Table of n, a(n) for n=1..35.


EXAMPLE

2^((32771)/2) == 1 (mod 3277), 3^((32811)/2) == 1 (mod 3281), ...


MATHEMATICA

residueQ[n_, m_] := Module[{ans = 0}, Do[If[Mod[k^2, m] == n, ans = True; Break[]], {k, 0, Floor[m/2]}]; ans]; A020649[n_] := Module[{m = 0}, While[ residueQ[m, n], m++]; m]; aQ[n_] := CompositeQ[n] && PowerMod[A020649[n], ((n  1)/2), n] == n  1; Select[Range[3, 110000, 2], aQ] (* Amiram Eldar, Apr 27 2019 *)


CROSSREFS

Cf. A001262, A006970, A020649, A047713, A053760, A244626, A307798 (the "residue" pseudoprimes), A307809.
Sequence in context: A237085 A260410 A223430 * A307809 A141629 A116460
Adjacent sequences: A307764 A307765 A307766 * A307768 A307769 A307770


KEYWORD

nonn


AUTHOR

Thomas Ordowski, Apr 27 2019


EXTENSIONS

More terms from Amiram Eldar, Apr 27 2019


STATUS

approved



