This is a computational problem, I don't know how much theoretical. With Pari, just for fun, I found: \begin{array}{c|c} 2 & 19\\ \hline 3 & 29\\ \hline 5 & 487\\ \hline 7 &22053404036884180513958627370176317218668443714432289646932414656004065291773402535727759 \\ \hline 11 & 33223 \end{array} The first column is the $n$-th prime $p_n$, whereas the second column is the prime $\lfloor p_n*\pi^k\rfloor=A(n)$ with the smallest possible $k$ positive integer, i mean k is the smallest positive integer for which $A(n)$ is prime. My question is: can exist two or more primes $A(n)$'s which are the same?
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$\begingroup$ @Brian I mean the smallest possible k for which A(n) is prime $\endgroup$ – user623145 Mar 22 '19 at 15:35
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$\begingroup$ Is the existence of k obvious? $\endgroup$ – Lior B-S Mar 22 '19 at 15:37
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$\begingroup$ @Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same. $\endgroup$ – user623145 Mar 22 '19 at 15:39
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$\begingroup$ These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky. $\endgroup$ – Lior B-S Mar 22 '19 at 15:40
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1$\begingroup$ @Lior B-S from nothing... $\endgroup$ – user623145 Mar 22 '19 at 15:45
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$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples $$\begin{array}{c|cc|cc} A(p_i)&p_1&k_1&p_2&k_2\\ \hline 30727&9781&1&991&3\\ 200579&20323&2&6469&3\\ 129499&41221&1&13121&2\\ 138283&44017&1&14011&2\\ 170063&54133&1&17231&2\\ 182617&58129&1&18503&2\\ 593429&60127&2&19139&3\\ 684091&69313&2&22063&3\\ 237689&75659&1&24083&2\\ 252869&80491&1&25621&2\\ 293729&93497&1&29761&2\\ \end{array}$$
answered Mar 22 '19 at 16:07
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$\begingroup$ any other solution? $\endgroup$ – user623145 Mar 22 '19 at 16:14
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$\begingroup$ 9781/991 is infact a good approximation of pi^2 $\endgroup$ – user623145 Mar 22 '19 at 16:17
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$\begingroup$ yes there are plenty, even without looking too hard. I'll edit my post $\endgroup$ – Thomas Lesgourgues Mar 25 '19 at 10:50
