A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

217, 13333, 16297, 23653, 30673, 44173, 46657, 48133, 56033, 98173, 130417, 131977, 136137, 179893, 188113, 190513, 197633, 267977, 334153, 334657, 347777, 360533, 407353, 412933, 421637, 486157, 667153, 670033, 677917, 694153, 710533, 765073, 839833, 935137, 997633

Offset

1,1

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 == 13, 17 (mod 20) such that 5^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

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)

Example

217 is a term because (5/217) = -1, and 5^((217-1)/2) == 1 (mod 217).

Prog

(PARI) isA375816(k) = (k>1) && gcd(k, 10)==1 && Mod(5, k)^((k-1)/2)==-kronecker(5, k)
(PARI) isA375816(k) = (k%20==13 || k%20==17) && Mod(5, k)^((k-1)/2)==1

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 | A375490 | this seq |

(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) | | | |

Sequence in context: A250448 A221427 A086026 * A100794 A048258 A373762

Adjacent sequences: A375813 A375814 A375815 * A375817 A375818 A375819

Keyword

nonn

Author

Jianing Song, Sep 01 2024

Status

approved