雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(49)
One question had been solved very early on.
Euclid had been able to show that there were infinitely many prime numbers, so that although in 1937 the number 2127 – 1 = 170141183460469231731687303715884105727 was the largest known prime, it was also known that they continued for ever.
也就是歐幾里德證明了素數有無限多個。所以，雖然在1937年人們已知的最大素數是（圖）2^127 - 1 = 170141183460469231731687303715884105727，但是人們知道永遠還有更大的。But another property that was easy to guess, but very hard to prove, was that the primes would always thin out,
然而，素數還有另一個性質，雖然很容易看出來，但卻很難證明，那就是它的分佈越來越稀疏。
at first almost every other number being prime, but near 100 only one in four, near 1000 only one in seven, and near 10,000,000,000 only one in 23.
一開始幾乎每個數都是素數，100以內則只有1/4的數是素數，1000以內只有1/7，而到了10, 000, 000, 000以內，就只有1/23了。There had to be a reason for it.
人們需要知道這是爲什麼。In about 1793, the fifteen-year-old Gauss noticed that there was a regular pattern to the thinning-out.
大約在1793年，15歲的高斯注意到，這個稀釋的過程是有規律的。The spacing of the primes near a number n was proportional to the number of digits in the number n;
n以內的的素數的間距，與n的大小有關，more precisely, it increased as the natural logarithm of n.
準確地說，它與n的自然對數成正比。Throughout his life Gauss, who apparently liked doing this sort of thing, gave idle leisure hours to identifying all the primes less than three million, verifying his observation as far as he could go.
高斯在他的餘生中，只要有空就去驗證這個猜想，他很喜歡做這樣的事。他檢查了3, 000, 000以內的所有素數，死而後已。
