A375917 Composite numbers k == 1, 11 (mod 12) such that 3^((k-1)/2) == 1 (mod k).

121, 1729, 2821, 7381, 8401, 10585, 15457, 15841, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 46657, 47197, 49141, 50881, 52633, 55969, 63973, 74593, 75361, 82513, 87913, 88573, 93961, 111361, 112141, 115921, 125665, 126217, 138481, 148417, 172081

Offset: 1

Author

Jianing Song, Sep 02 2024

Keywords

• nonn

Comments

Odd composite numbers k such that 3^((k-1)/2) == (3/k) = 1 (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol).

It seems that most terms are congruent to 1 modulo 12. The first terms congruent to 11 modulo 12 are 1683683, 1898999, 2586083, 2795519, 4042403, 4099439, 5087171, 8243111, ...

Examples

			1683683 is a term because 1683683 = 59*28537 is composite, 1683683 == 11 (mod 12), and 3^((1683683-1)/2) == 1 (mod 1683683).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..1000

Crossrefs

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

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

(b/k)=1, b^((k-1)/2)==1 (mod k) | A006971 | this seq | 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 | A375490 | 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

isA375917(k) = (k>1) && !isprime(k) && (k%12==1 || k%12==11) && Mod(3,k)^((k-1)/2) == 1