Hello all, Dirichlet's famous theorem asserts that any arithmetic progression
$\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a
and b are relatively prime.
I am wondering if the following strengthening of Dirichlet's theorem is also
true :
Let $a,b$ be relatively prime integers
as above. Then there is a uniform bound $g(a,b)$ such
that any interval $\lbrace x+1,x+2, \ldots ,x+g(a,b)\rbrace$ of $g(a,b)$
successive integers contains at least one integer $y$ which is
congruent to $b$ modulo $a$ and which is not divisible by
any integer between $x+1$ (inclusive if $y\neq x+1$) and
$y$ (exclusive).
Without the uniform bound, this would be a tasteless easy consequence
of Dirichlet's theorem. With the bound, however, it becomes stronger
than Dirichlet's theorem.
Perhaps the two are in fact equivalent ?
Best Answer
This question has essentially been answered in the comments, so I see no reason for it to remain officially unanswered.
The proposed property holds. It is "equivalent to" Dirichlet's theorem, in the informal sense that each could be used to prove the other. However, we are not talking about axiomatization, so rather I would say that both are true and equivalence isn't meaningful. (This is not to be confused with Fréchet's objection to Tarski.)
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