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Is there any proof of infinitude of A007500 primes?

If you want to generate them here is trivial and naive python program.

def is_prime(n):
    i = 2
    while i*i <= n:
        if n%i == 0:
            return False
        i = i + 1
    else:
        return True

print [x for x in range(1,200) if is_prime(x) and is_prime(int(str(x)[::-1]))]
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    $\begingroup$ Mark Sapir: that page is about a subset of the set OP asks about, and does not say whether that subset is infinite. Thus, I don't see how this answers the question. $\endgroup$
    – Zsbán Ambrus
    Commented Dec 9, 2011 at 9:49
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    $\begingroup$ It says that "the largest known ... is ...". What else do you want? All such problems are open. The maximal result about a natural set of numbers with infinite subset of primes is the Dirichlet theorem. Everybody with access to Google can learn it in a few minutes. $\endgroup$
    – user6976
    Commented Dec 9, 2011 at 10:01
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    $\begingroup$ The OP is not asking about palindromes which are primes, and that is why the linked mathworld page is irrelevant. The OP is asking about primes whose reverses are also primes, regardless of whether the number and its reverse are the same. $\endgroup$
    – Sridhar Ramesh
    Commented Dec 9, 2011 at 11:01
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    $\begingroup$ @Sridhar Ramesh: These questions are the same: this does not follow from Dirichlet (or Chebotarev density), hence the answer is unknown. One can easily cook up infinitely many questions like that. For example: are there infinitely many primes of the form $x^2+1$ ($x^2+2$, $x^2+3, $2^n+1, 2^n+3$, etc.). $\endgroup$
    – user6976
    Commented Dec 9, 2011 at 13:04
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    $\begingroup$ @Mark Sapir: while I agree with the general sentiment, that there are inifnitely many primes of the form x^2 + y^4 does not follow from Dirichlet but is still known; also there are some results on primes with given Hamming weight and some other things. $\endgroup$
    – user9072
    Commented Dec 9, 2011 at 15:19

3 Answers 3

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The answer is: no proof is known at this time, for any base, but it is suspected that a proof exists (it should be reasonably easy to give a density under the "standard hypotheses").

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Now, thinking about this a bit, let's say $f$ is the function that reverses the digits, so that $f(n)$ is the number that has the digits of $n$ in base 10 reversed. I think that when estimating $$|\{n \leq x: n,f(n) \mbox{ simultaneously prime}\}|,$$

then for each prime $p$, maybe you'll have to estimate the number of solutions to $$ nf(n) \equiv 0 \bmod p, $$ where $n \in \mathbb{Z}/p\mathbb{Z}$. This is similar to like when estimating the twin prime constant (but I'm not claiming that the whole thing goes through the same way). The problem is that $f(n)$ is not so straightforward like $n+2$ is. For $p=3$, at least $f(n) \equiv n \bmod 3$, so that is ok. $p=11$ is also not too bad. For other $p$, it doesn't seem so straightforward.

In fact, it's not even as straightforward as this, because for $n,f(n)\in \mathbb{Z}$, when one fixes $n \bmod p$, $f(n) \bmod p$ is not fixed in general. The point is just that I think one might have to estimate the probability that $p\nmid f(n)\in \mathbb{Z}$, given that $p\nmid n\in \mathbb{Z}$.

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Hello all,

I must be overlooking something, but I wonder if the systems $\Psi:\mathbb{Z}^d\rightarrow \mathbb{Z}^t$ in the Green-Tao paper "Linear Equations in Primes" could apply to this question.

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