A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

1541, 2465, 4961, 30857, 31697, 72041, 83333, 162401, 192713, 206981, 258017, 359369, 544541, 565001, 574397, 653333, 929633, 1018601, 1032533, 1133441, 1351601, 1373633, 1904033, 1953281, 2035661, 2797349, 2864501, 3264797, 3375041, 3554633, 3562361, 3636161

Offset: 1

Author

Jianing Song, Sep 01 2024

Keywords

• nonn

Comments

Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 5 (mod 12) such that 3^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

Examples

			1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..1000
• Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

Crossrefs

| b=2 | b=3 | b=5 |

-----------------------------------+-------------------+----------+---------+

(b/k)=1, b^((k-1)/2)==1 (mod k) | A006971 | A375917 | A375915 |

-----------------------------------+-------------------+----------+---------+

(b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916 |

-----------------------------------+-------------------+----------+---------+

b^((k-1)/2)==-(b/k) (mod k), also | A306310 | this seq | A375816 |

(b/k)=-1, b^((k-1)/2)==1 (mod k) | | | |

-----------------------------------+-------------------+----------+---------+

Euler-Jacobi pseudoprimes | A047713 | A048950 | A375914 |

(union of first two) | | | |

-----------------------------------+-------------------+----------+---------+

Euler pseudoprimes | A006970 | A262051 | A262052 |

(union of all three) | | | |

Programs

PARI

isA375490(k) = (k>1) && gcd(k,6)==1 && Mod(3,k)^((k-1)/2)==-kronecker(3,k)

PARI

isA375490(k) = k%12==5 && Mod(3,k)^((k-1)/2)==1 \\ Jianing Song, Sep 07 2024