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In this paper: Iwaniec and Urroz - Orders of CM elliptic curves modulo $p$ with at most two primes on page 818, the authors claimed in the second paragraph that under some conditions, the statement of Theorem 1.2 in page 817 is equivalent to the twin primes conjecture. I am searching for this result, but without any success. Then I am asking for a proof or a reference where I can find this proof.

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  • $\begingroup$ The paper actually uses the phrase "Twin Prime Conjecture", which I think has a different meaning. This would be something like: fix a and b to be nonzero and coprime integers of different parity, then there are infinitely many twins p,q, where p and q are both prime and satisfy p=aq + b. But I'm not an expert, so search the literature for more uses of "Twin Prime Conjecture". Gerhard "Often Engages In Double Meaning" Paseman, 2020.01.01. $\endgroup$ Jan 2, 2020 at 7:52
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    $\begingroup$ You have misunderstood what they wrote. Note the start of the second paragraph indicates it is about primes $p \equiv 3 \bmod 4$, and Theorem 1.2 is about primes $p \equiv 1 \bmod 4$, so they're not about the same primes. What the authors say is "essentially equivalent" to the twin prime conjecture is that there should be infinitely many primes $q$ for which $4q-1$ is also prime. Asking if two polynomials $x$ and $ax+b$ can both take prime values at the same time infinitely often sounds like the twin prime conjecture (the special case of $x$ and $x+2$). I think that's all they meant. $\endgroup$
    – KConrad
    Jan 2, 2020 at 8:06
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    $\begingroup$ If that's what you are asking about, Helena, then that's what it should say in the body of your question. Please edit the question so it asks what you actually mean to ask. $\endgroup$ Jan 2, 2020 at 14:53
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    $\begingroup$ The conjectural growth of twin primes can be connected to the conjectural behavior of certain Dirichlet series as $s \rightarrow 1$, but those series are not known to have an analytic continuation outside the half-plane ${\rm Re}(s) > 1$. See Theorem 11 in kconrad.math.uconn.edu/articles/hlconst.pdf, for instance. In the 1st chapter of "Math Talks for Undergraduates" (есть русский перевод: «Математические беседы для студентов») Serge Lang wrote: "I regard it as a major problem to give an estimate for the error term in the Bateman-Horn conjecture similar to the Riemann hypothesis." $\endgroup$
    – KConrad
    Jan 2, 2020 at 17:48
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    $\begingroup$ The same Chen switch technique that gives $p-2=P_2$ gives $4p-1=P_2$. Any sieve that would give you $4p-1$ is prime infinitely often could be appropriately modified to give you $p-2$ is prime infinitely often. So, these problems are equally hard, and solving them would ultimately mean cracking the Parity problem for sieves. $\endgroup$ Jan 2, 2020 at 19:31

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