OKlibrary  0.2.1.6
general.hpp File Reference

Investigating the transversal hypergraph of hypergraphs of arithmetic progressions in the primes. More...

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Detailed Description

Investigating the transversal hypergraph of hypergraphs of arithmetic progressions in the primes.

Similar to Investigations/RamseyTheory/VanderWaerdenProblems/Transversals/plans/general.hpp, but now for given n we do not consider the set {1,...,n}, but the set of the first n prime numbers.

Todo:
Basic considerations
  • Let tau_arithprog_primes(k,n) be the transversal number of the hypergraph arithprog_primes_hg(k,n) with vertex set the first n prime numbers and the arithmetic progressions of size k in that set as hyperedges.
  • The Green-Tao theorem states that for every k >= 1 the limit of tau_arithprog_primes(k,n)/n is 1.
  • Let the transversal hypergraph of arithprog_primes_hg be called tr_arithprog_hg(k,n).
Todo:
Investigating greentao_2(2,k)
  • Simple function for computing greentao_2(2,k):
    greentao2(k) := block([break:false,res],
     for n : 0 thru inf unless break do block([G : arithprog_primes_hg(k,n)[2]],
     if not emptyp(G) and emptyp(lintersection(G)) then (break:true,res:n)),
     res)$
       
  • Easy to compute for k <= 10:
    for k : 1 thru 10 do print(k, greentao2(k));
        1  2
        2  3
        3  7
        4  14
        5  31
        6  55
        7  214
        8  349
        9  690
       10  5482
       

Definition in file general.hpp.