Browkin猜想
Browkin's discriminator conjecture
摘要：
设$q\ge 5$是素数，并将$q^*=(-1)^{(q-1)/2}\cdot q$。考虑整数序列$u_q(1),u_q(2),\ldots,$与$u_q(j)=(3^j-q^*(-1)^j)/4$ .该序列中没有重复的项，因此对于每个$n$，存在最小整数$m$，使得$u_q(1),\ldots,u_q(n)$以$m$为模成对不全等。我们写$D_q(n)=m$。考虑$D_q(n)$的想法是由于Browkin（2015），他在$3$是模$q,$的原根的情况下证明了$D_q(n)$假设的唯一值是$2$和$q$的幂。我们表明，这是真实的$n\neq 5$，但虚假的情况下$n=5$无限多$q$。我们还确定情况3中的$D_q(n)$不是模$q$的原根。Browkin猜想的灵感来自Moree和Zumalacárregui（2016）的早期工作，他们确定了$D_5(n)$为$n\ge 1$，从而建立了Salajan猜想。对于一个固定的素数$q$，他们的方法很容易推广，但需要一些创新，以处理所有素数$q\ge 7$和所有$n\ge 1$。有趣的是，费马素数和米里曼诺夫素数在其中扮演了一个特殊的角色。
英文摘要：
Let $q\ge 5$ be a prime and put $q^*=(-1)^{(q-1)/2}\cdot q$. We consider the integer sequence $u_q(1),u_q(2),\ldots,$ with $u_q(j)=(3^j-q^*(-1)^j)/4$. No term in this sequence is repeated and thus for each $n$ there is a smallest integer $m$ such that $u_q(1),\ldots,u_q(n)$ are pairwise incongruent modulo $m$. We write $D_q(n)=m$. The idea of considering the discriminator $D_q(n)$ is due to Browkin (2015) who, in case $3$ is a primitive root modulo $q,$ conjectured that the only values assumed by $D_q(n)$ are powers of $2$ and of $q$. We show that this is true for $n\neq 5$, but false for infinitely many $q$ in case $n=5$. We also determine $D_q(n)$ in case 3 is not a primitive root modulo $q$. Browkin's inspiration for his conjecture came from earlier work of Moree and Zumalacárregui (2016), who determined $D_5(n)$ for $n\ge 1$, thus establishing a conjecture of Salajan. For a fixed prime $q$ their approach is easily generalized, but requires some innovations in order to deal with all primes $q\ge 7$ and all $n\ge 1$. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
