cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A192579 Primes p for which there is no prime q == 3 (mod 4) that is smaller than p and is a quadratic residue modulo p.

Original entry on oeis.org

2, 3, 5, 7, 17
Offset: 1

Views

Author

Jonathan Sondow, Jul 04 2011

Keywords

Comments

Gica proved that if p is a prime different from 2, 3, 5, 7, 17, then there exists a prime q < p which is a quadratic residue modulo p and q == 3 (mod 4).
This is the unique set of primes answering the question in the Mathematics Stack Exchange link. - Rick L. Shepherd, May 29 2016

Examples

			p = 17 is a member, because the primes q < p with q == 3 (mod 4) are q = 3, 7, 11, and they are not quadratic residues modulo 17.
11 is not a member, because 3 < 11 and 3 == 5^2 (mod 11).

Links

Crossrefs

Cf. A192578.

A193063 Primes p for which there is no prime q == 1 (mod 3) that is smaller than p and is a quadratic residue modulo p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13
Offset: 1

Views

Author

Jonathan Sondow, Jul 15 2011

Keywords

Comments

Gica proved that if p is a prime different from 2, 3, 5, 7, 11, 13, then a prime q < p exists which is a quadratic residue modulo p and q == 1 (mod 3). His paper has not yet been published, but see A192578 for a reference, link, and examples of a similar result.

Crossrefs

Cf. A192578, A192579, A193064.

Programs

  • Mathematica
    piMax = 100; quadQ[q_, p_] := Length[ Select[ Table[x^2, {x, 1, Floor[p/2]}], Mod[#, p] == q & , 1]] == 1; noPrimeQ[p_] := Select[ Table[ Prime[n], {n, 1, PrimePi[p]}], Mod[#, 3] == 1 && quadQ[#, p] &] === {}; Select[ Table[ Prime[n], {n, 1, piMax}], noPrimeQ] (* Jean-François Alcover, Oct 03 2011 *)

A193064 Primes p for which there is no prime q == 2 (mod 3) that is smaller than p and is a quadratic residue modulo p.

Original entry on oeis.org

2, 3, 5, 13
Offset: 1

Views

Author

Jonathan Sondow, Jul 15 2011

Keywords

Comments

Gica proved that if p is a prime different from 2, 3, 5, 13, then a prime q < p exists which is a quadratic residue modulo p and q == 2 (mod 3). His paper has not yet been published, but see A192578 for a reference, link, and examples of a similar result.

Crossrefs

Cf. A192578, A192579, A193063.
Showing 1-3 of 3 results.

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