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A140947 Four-columned array read by rows: each row gives a series of 4 consecutive primes that share a 2nd-degree polynomial relationship and produce a positive-only integer series from the derived quadratic.

%I A140947 #3 Mar 31 2012 10:32:52
%S A140947 17,19,23,29,41,43,47,53,79,83,89,97,227,229,233,239,347,349,353,359,
%T A140947 349,353,359,367,379,383,389,397,439,443,449,457,569,571,577,587,641,
%U A140947 643,647,653,673,677,683,691,677,683,691,701,1031,1033,1039,1049
%N A140947 Four-columned array read by rows: each row gives a series of 4 consecutive primes that share a 2nd-degree polynomial relationship and produce a positive-only integer series from the derived quadratic.
%C A140947 These "proximate-prime polynomials" exhibit high prime densities. Of the 333 under 100000, 46 have greater than 50% prime values for the first 1000 terms. 2221 positive-only PPPs have been found under 1000000. All positive-integer PPPs have complex roots (only negative-integer PPPs, which are excluded) have real roots. The roots mostly have a real part of 1/2 or a multiple of 1/2.
%D A140947 Purple Math: Finding the Next Number in a Sequence: The Method of Common Differences http://www.purplemath.com/modules/nextnumb.htm
%D A140947 Robert Sacks, Method of Common Differences http://www.numberspiral.com/p/common_diff.html
%H A140947 Michael M. Ross <a href="http://www.naturalnumbers.org/ppanalysis.html">The High Primality of Prime-Derived Quadratic Sequences (2007)</a>
%H A140947 Michael M. Ross <a href="http://www.naturalnumbers.org/QTest-NTK.html">How to Use Qtest (2007)</a>
%F A140947 Method of common differences: if (P2 - P1) - (P3 - P2) = (P3 - P2) - (P4 - P3) then polynomial is degree 2.
%e A140947 For 17, 19, 23, 29 the method of common differences produces coefficients of 1, -1 and 17 for a polynomial expression of n^2 - n + 17.
%Y A140947 Cf. A126665, A126719.
%K A140947 nonn,uned,tabf
%O A140947 1,1
%A A140947 _Michael M. Ross_, Jul 24 2008