bibkey: smith2017a136162
authors: Harry J. Smith; Jaroslav Krizek
year: 2017
title: "OEIS A136162, List of prime quadruplets {p, p+2, p+6, p+8}"
doi: null
url: https://oeis.org/A136162
claim: "%N List of prime quadruplets {p, p+2, p+6, p+8}. %C {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q*(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - _Jaroslav Krizek_, Jul 07 2017"
strata_touched:
  - D5/S3/Arith/Congruence/KrizekNextPrimeProductQuadruplet
license: citation-only
triage: anchor

OEIS A136162

The NAME of A136162 states:

List of prime quadruplets {p, p+2, p+6, p+8}.

The Krizek COMMENT contains two statements:

{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12k^2-1) and 3(12k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - Jaroslav Krizek, Jul 07 2017

The AUTHOR line is:

Harry J. Smith, Dec 17 2007

Harry J. Smith is the sequence author, while Jaroslav Krizek authored the 2017 conjecture. The year records the conjecture date rather than the 2007 sequence-author date.

The formal result attached to this note proves only the second, nextprime-product sentence. The twin-prime-product sentence is not claimed here and is proved in the OEIS entry itself by the displayed factors 3*(12*k^2-1) and 3*(12*k^2+1).

Verified locator

• URL: https://oeis.org/A136162
• NAME (verbatim): List of prime quadruplets {p, p+2, p+6, p+8}.
• COMMENT (verbatim): {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12k^2-1) and 3(12k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - Jaroslav Krizek, Jul 07 2017
• AUTHOR (verbatim): Harry J. Smith, Dec 17 2007