To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician Jakob Kulik had embarked on an ambitious project to find all the primes up to 100 million.

　　为了自动化冗长乏味的筛分步骤，德国数学家 Carl Friedrich Hindenburg 用可调节的滑动条在整页表格上一次排除所有倍数。另一种技术含量低但非常有效的方法是用漏字板来查找倍数的位置。到了19世纪中叶，数学家 Jakob Kulik 开始了一项雄心勃勃的计划，他要找出1亿以内的所有质数。

　　This “big data” of the 1800s might have only served as reference table, if Carl Friedrich Gauss hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “chiliad,” or group of 1000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on.

　　若没有高斯等人对质数的研究，这个19世纪的“大数据”或许只能作为一张参考表。在有了这张包含300万以内所有质数的列表之后，高斯开始着手数它们，每次以1000为分界点分组。他找出1000以内的质数，然后再找出1000到2000之间的质数，然后是2000到3000之间，以此类推。

　　Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an “inverse-log” law. Gauss’s law doesn’t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The correct count is 75 primes, about a 4 percent error.

　　高斯发现，随着数值的增高，质数出现的频率会遵循“反对数”定律逐渐下降。虽然高斯定律没确切地给出质数的数量，但它给出了一个非常好的估计。例如他预测了从1,000,000至1,001,000之间大约有72个质数;而正确的计数是75个，误差值约为4%。

　　A century after Gauss’ first explorations, his law was proved in the “prime number theorem.” The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, a million-dollar prize problem today, also describes how accurate Gauss’ estimate really is.

　　在高斯的第一次探索之后的一个世纪里，他的定律在“质数定理”中得到了证明。在数值越大的质数范围内，它的误差百分比接近于零。作为世界七大数学难题之一的黎曼假设，也描述了高斯估算的准确程度。

　　The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.

　　质数定理和黎曼假设都得到了应有的关注和资金，但这两者都是在早期不那么迷人的数据分析中得到的。

　　Modern prime mysteries

　　现代质数之谜

　　Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.

　　现在，我们的数据集来自计算机程序而非手工切割的漏字模板，但数学家仍在努力寻找质数中的新模式。

　　Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.

　　除了2和5之外，所有质数都以数字1、3、7、9结尾。在19世纪，数学家证明了这些可能的结尾数字有着同样的出现频率。 换句话说，如果数100万以内的质数，会发现大约25%的质数以1结尾，25%以3结尾，25%以7结尾，以及25%以9结尾。

　　A few years ago, Stanford number theorists Robert Lemke Oliver and Kannan Soundararajan were caught off guard by quirks in the final digits of primes. An experiment looked at the last digit of a prime, as well as the last digit of the very next prime. For example, the next prime after 23 is 29: One sees a 3 and then a 9 in their last digits. Does one see 3 then 9 more often than 3 then 7, among the last digits of primes?

　　几年前，斯坦福大学的数论学家 Robert Lemke Oliver 和 Kannan Soundararajan 在一个观察质数和下一个质数的最后一位数字的实验中，发现了质数的结尾数的奇异之处。例如质数23之后的下一个质数是29，它们的结尾数字分别是3和9。那么是否在质数的结尾数中，3和9的出现要多过于3和7吗?

　　Number theorists expected some variation, but what they found far exceeded expectations. Primes are separated by different gaps; for example, 23 is six numbers away from 29. But 3-then-9 primes like 23 and 29 are far more common than 7-then-3 primes, even though both come from a gap of six.

　　数论学家预计会有一些变化，但他们的发现远远超出预期。质数与质数之间被不同大小的间隔分开;例如，23与29之间相差6。但是像23和29那样的先以3再以9结尾的质数比先以7再以3结尾的质数要普遍得多，尽管这两种质数组合的间隔都是6。

　　Mathematicians soon found a plausible explanation. But, when it comes to the study of successive primes, mathematicians are (mostly) limited to data analysis and persuasion. Proofs – mathematicians’ gold standard for explaining why things are true – seem decades away.

　　虽然数学家很快找到了合理的解释。但是，在研究连续质数时，数学家大多能做的仅限于数据分析和尽力说服。而数学家用以解释某事物为何为真的黄金标准——证明，似乎仍距我们数十年之远。