Prime Generating Polynomials Across multiple gp runs, similar error values continue to emerge which prevent progress. To identify the values, take the push program produced and run it in inter.ss. The values will be in reverse order from the errors in the gp run. Thus, if an error is 9 values from the end, then the value from inter.ss is 9 values from the beginning [of the sequence]. Values such as: Error Values Product of Primes 10/41 1681 41,41 830/1763 1763 41,43 890/2021 2021 43,47 These values are products of primes. They are also the consecutive non-prime values on an ulam spiral starting at 41. [http://www.algorithm.co.il/blogs/index.php/tag/ulam-spiral/] Possibilities moving forward / modifications: - research products of primes - effect on totient values - heuristic for fitness function (modification) - figure out fix - regression to sequence of primes? - make multivariable? - prime or uniquely expressible by prime factors produced by another polynomial http://www.mathpages.com/home/kmath058.htm - evolve second polynomial? - proximate prime polynomials? http://www.naturalnumbers.org/ http://www.naturalnumbers.org/highpolys.html http://www.naturalnumbers.org/polyalignment.html - percentage of primes included in fitness function - distinct primes - sections of ordered primes - evolve prime-poly of form L(n)=an+b (Dirichlet's theorem on arithmetic progressions) - evolve higher prime generating polynomials based on Green-Tao theorem - for any k there exists a pair of a and b with the property that L(n) = an + b is prime for any n from 0 to k-1. - best known result of such type is for k = 26 -Term stack term 1: 3x^2 term 2: -5x term 3: +4 each term on the stack has the following attributes: sign (+/-) coefficient variable (x) {constant for single variable} exponent (integer n, where n occurs as x^n) - prime generating power series? - Lagrange polynomial on multiple gp run results or preexisting prime generating polynomials? - It is not known whether there exists a univariate polynomial of degree at least 2 that assumes an infinite number of values that are prime. - from wikipedia formula for primes page Notes on numbers 1681: 41^2, smallest number yielded by the formula n^2 + n + 41 that is not a prime; centered octagonal number 1763: Divisors of the positive integer 1763 1,41,43,1763 Links on numbers: http://answers.yahoo.com/question/index?qid=20080731102004AAgolke http://www.positiveintegers.org/1763 http://answers.yahoo.com/question/index?qid=20100301012458AA58j85 http://hej.sze.hu/ANM/ANM-000926-A/anm000926a/node3.html Papers to consult?: 1947 H. Mills proved that there exists a real number A, for which A^3^n is always prime for an arbitrary positive integer n. external links on wikipedia formula for primes page (pdfs) Ideas to consult: Ulam Spiral : http://en.wikipedia.org/wiki/Ulam_spiral http://en.wikipedia.org/wiki/Polynomial http://en.wikipedia.org/wiki/Formula_for_primes Heegner number [wiki] http://www.maa.org/editorial/mathgames/mathgames_07_17_06.html Liouville function moebius function Contests: http://www.recmath.org/contest/PGP/index.php Threads http://docs.plt-scheme.org/reference/eval-model.html#%28part._thread-model%29 something random General outline for presentation: - initial generations - generating for values less than range 41 - return good functions completely prime - increase range to 50 - gp's failing - similar errors - translated error values into regular values, similar values occuring across runs values are products of primes > 41 all values are on downward diagonal of ulam spiral beginning at 41 [ explain why prime polynomial cannot generate only primes ] [ explain ulam spiral ] (why originally began at 41?)