Proof of infinitude of primes whose reversal in base 10 is also prime - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T10:51:09Z http://mathoverflow.net/feeds/question/83044 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83044/proof-of-infinitude-of-primes-whose-reversal-in-base-10-is-also-prime Proof of infinitude of primes whose reversal in base 10 is also prime Pratik Deoghare 2011-12-09T09:19:35Z 2011-12-09T14:34:54Z <p>Is there any proof of infinitude of <a href="http://oeis.org/A007500" rel="nofollow">A007500</a> primes?</p> <p>If you want to generate them here is trivial and naive python program.</p> <pre><code>def is_prime(n): i = 2 while i*i &lt;= 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]))] </code></pre> http://mathoverflow.net/questions/83044/proof-of-infinitude-of-primes-whose-reversal-in-base-10-is-also-prime/83049#83049 Answer by Igor Rivin for Proof of infinitude of primes whose reversal in base 10 is also prime Igor Rivin 2011-12-09T10:54:26Z 2011-12-09T10:54:26Z <p>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").</p> http://mathoverflow.net/questions/83044/proof-of-infinitude-of-primes-whose-reversal-in-base-10-is-also-prime/83050#83050 Answer by Timothy Foo for Proof of infinitude of primes whose reversal in base 10 is also prime Timothy Foo 2011-12-09T11:24:15Z 2011-12-09T11:33:19Z <p>Hello all,</p> <p>I must be overlooking something, but I wonder if the systems $\Psi:\mathbb{Z}^d\rightarrow \mathbb{Z}^t$ in the Green-Tao paper <a href="http://arxiv.org/abs/math/0606088" rel="nofollow">"Linear Equations in Primes"</a> could apply to this question. </p> http://mathoverflow.net/questions/83044/proof-of-infinitude-of-primes-whose-reversal-in-base-10-is-also-prime/83057#83057 Answer by Timothy Foo for Proof of infinitude of primes whose reversal in base 10 is also prime Timothy Foo 2011-12-09T14:00:40Z 2011-12-09T14:34:54Z <p>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 <code>$$|\{n \leq x: n,f(n) \mbox{ simultaneously prime}\}|,$$</code></p> <p>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.</p> <p>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 $p\nmid n\in \mathbb{Z}$.</p>