Number Theory

    

Thick Sequences

Authors: Joseph Pe

We call an integer sequence thick if the quotients formed from its terms are dense in the set of real numbers. To find thick sequences, we consider the geometric, Fibonacci, power, and prime sequences. We show that the sequence of primes is thick provided that a conjecture DC-2 holds. DC-2 says that certain pairs of linear Dirichlet conditions have infinitely many solutions. It is a weak form of Dickson’s conjecture, which states that a finite system of linear Dirichlet conditions has infinitely many solutions and generalizes Dirichlet’s well-known result on primes in arithmetic progressions. Also, we obtain partial results for the general thickness problem for an arbitrary sequence and look at heuristic evidence for the validity of DC-2. We conclude with a short list of problems for further research.

Comments: 7 Pages.

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[v1] 2025-03-13 02:38:18

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