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  • Sum of Digits is Prime

    Sum of Digits is Prime

    August 19, 2014

    For an integer $qgeqslant2$ let $s_q(n)$ denote the $q$-ary sum-of-digits function of a non-negative integer $n$, that is, if $n$ is given by its $q$-ary digits expansion $n=sumlimits_{k=0}^{r} a_k q^{k}$ with digits $a_kin{0,1,ldots,q-1}$ and $a_r eq0$, then
    [s_q(n)=sum_{k=0}^{r} a_k.]

    As usual we write $phi(n)$ for Euler's totient function, and $pi(x)$ for the number of primes up to $x$. We recall the prime    number theorem in the form

    egin{equation}pi(x)=frac{x}{log x}+Oleft(frac{x}{(log x)^2} ight).end{equation}

    Theorem (Drmota/Mauduit/Rivat)    We have uniformly for all integers $kgeqslant 0$ with $(k,q-1)=1$,

    egin{equation}label{eq:2}#{pleqslant x:s_q(p)=k}=frac{q-1}{phi(q)}frac{pi(x)}{sqrt{22pisigma_{q}^{2}log_q x}}left(expleft(-frac{(k-mu_qlog_q x)^2}{2sigma_{q}^{2}log_q x} ight)+Oig((log x)^{-1/2+varepsilon}ig) ight),end{equation}

    where $varepsilon>0$ is arbitrary but fixed, and $mu_q:=frac{q-1}{2}$, $sigma_{q}^{2}:=frac{q^2-1}{12}$.

    From the above result we can deduce some very interesting corollaries. Clearly, eqref{eq:2} shows that every sufficiently large integer is the sum of digits of a prime number. So, in particular, there are infinitely many primes whose sum of digits is   also prime. Also, every sufficiently large prime is the sum of digits of another prime which in turn is the sum of digits of another prime, and so on.

    References

    1. C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. 171 (2010), 1591–1646.
    2. Glyn Harman, Counting Primes whose Sum of Digits is Prime, Journal of Integer Sequences, Vol. 15 (2012), Article 12.2.2.
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  • 原文地址:https://www.cnblogs.com/pengdaoyi/p/4279375.html
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