Search: id:A134969
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%I A134969
%S A134969 3,11,5,17,7,19,13,29,17,37,23,43,29,53,43,71,67,103,71,107,73,109,97,
%T A134969 137
%N A134969 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 A134969 Michael M. Ross <a href="http://www.naturalnumbers.org/quadprimes.html">
               What Are Odd Squared, or Biquadratic, Paired Primes?</a>
%H A134969 Diagram of paired primes <a href="http://www.naturalnumbers.org/images/
               psline.gif">The diagram shows how two perfect pairs partially overlap: 
               23 and 43 span perfect squares 25 and 36; 29 and 53 span perfect 
               squares 36 and 49.</a>
%H A134969 VBA source code<a href="http://www.naturalnumbers.org/quadprimes.html#visualizing">
               Provides an Excel visualization of the odd-offset prime pairs and 
               colocation with twin primes.</a>
%F A134969 PS1 + OfN = P1, PS3 + OfN = P2 where PS1 = 1st perfect square, PS3 = 
               3rd perfect square OfN = an equal positive offset from preceding 
               perfect square P1 and P2 = the prime pair
%e A134969 For example, prime pair 97 and 137 have an even offset of 16 from 81 
               and 121:
%e A134969 Interval 1: 9*9 = 81 + 16 = 97
%e A134969 Interval 2: 10*10 = 100 + 16 = 116
%e A134969 Interval 3: 11*11 = 121 + 16 = 137
%e A134969 Prime pair 71 and 107 have an odd offset of 7 from 64 and 100:
%e A134969 Interval 1: 8*8 = 64 + 7 = 71
%e A134969 Interval 2: 9*9 = 81 + 7 = 88
%e A134969 Interval 3: 10*10 = 100 + 7 = 107
%e A134969 In all cases there is a complete intermediate quadratic interval (#2).
%o A134969 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 A134969 Sequence in context: A084466 A084462 A156320 this_sequence A139686 A130537 
               A114234
%Y A134969 Adjacent sequences: A134966 A134967 A134968 this_sequence A134970 A134971 
               A134972
%K A134969 nonn,tabf
%O A134969 1,1
%A A134969 Michael M. Ross (mr(AT)wordwisesolutions.com), Feb 04 2008

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