|
%I #20 Jul 12 2023 11:02:48
%S 3,11,5,17,7,19,13,29,17,37,23,43,29,53,43,71,67,103,71,107,73,109,97,
%T 137
%N List of pairs of primes that are separated by the equivalent of 2 quadratic intervals. Both primes are greater than their preceding perfect squares by the same amount, or offset. The respective perfect squares can be both odd, in which case the offset is even, or both even, in which case the offset is odd.
%H Michael M. Ross, <a href="http://www.naturalnumbers.org/quadprimes.html">What Are Odd Squared, or Biquadratic, Paired Primes?</a>
%H Michael M. Ross, <a href="http://www.naturalnumbers.org/images/psline.gif">Diagram of paired primes</a> showing how two perfect pairs partially overlap: 23 and 43 span perfect squares 25 and 36; 29 and 53 span perfect squares 36 and 49.
%H Michael M. Ross, <a href="http://www.naturalnumbers.org/quadprimes.html#visualizing">VBA source code</a> providing an Excel visualization of the odd-offset prime pairs and colocation with twin primes.
%F PS1 + OfN = P1, PS3 + OfN = P2, where
%F PS1 = first perfect square,
%F PS3 = 3rd perfect square,
%F OfN = an equal positive offset from preceding perfect square, and
%F P1 and P2 = the prime pair.
%e Prime pair 71 and 107 have an odd offset of 7 from 64 and 100:
%e Interval 1: 8*8 = 64 + 7 = 71.
%e Interval 2: 9*9 = 81 + 7 = 88.
%e Interval 3: 10*10 = 100 + 7 = 107.
%e Prime pair 97 and 137 have an even offset of 16 from 81 and 121:
%e Interval 1: 9*9 = 81 + 16 = 97.
%e Interval 2: 10*10 = 100 + 16 = 116.
%e Interval 3: 11*11 = 121 + 16 = 137.
%e In all cases there is a complete intermediate quadratic interval (#2).
%e The pair (11,83), 11-9 = 83-81 = 2, does not work because there are 6 squares in between: 16,25,36,49,64,81.
%o Excel VBA available at http://www.naturalnumbers.org/quadprimes.html#code Demonstrates odd-offset prime-pair distribution with associated twin primes. Limitations: -Only the subset of paired primes for even perfect squares with odd offsets are demonstrated. -Because of the 256-column limitation for an Excel worksheet, data in rows after perfect square 484 are not complete. -Only twin primes that are also paired primes are outlined (in red).
%Y Cf. A056892.
%K nonn,tabf,more,less
%O 1,1
%A _Michael M. Ross_, Feb 04 2008
| 