Green-Tao Theorem?

Number Theory Level 3

Does there exist an increasing sequence $t_1, t_2, \ldots$ such that for any integer constant $c$ , the sequence

$t_1+c, t_2+c, t_3+c, \ldots$

contains only a finite number of primes?

Yes No

2 solutions

Sharky Kesa
Mar 13, 2016

A simple construction could be inspired by thinking that you want $t_n$ to be eventually divisible by $c$ . With this in mind, we could try the sequence $t_n = n!$ . However, if we have $c=\pm 1$ , the sequence, may still contain infinite primes. However, we know that $a^3 \pm 1$ is factorable. Thus, we could try the sequence $t_n = (n!)^3$ , which we would find to satisfy having finite primes.

Efren Medallo
Sep 15, 2016

A simple example would be $t_n = 2n-1$ and $c=1$ . Since all $t_n + c$ will be even, then only the number $2$ will be prime.

This is not a correct solution, since I have said for any integer constant $c$ , not for a specific constant. In your case, if $c=2$ , there will be infinitely many primes that satisfy.

Sharky Kesa - 4 years, 9 months ago

Oh! Didn't notice that $c$ may assume any value. Sorry for that!

Efren Medallo - 4 years, 9 months ago

Green-Tao Theorem?

2 solutions

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