插入/从哈希表中检索时要做的第一件事是计算给定键的hashCode，然后通过执行hashCode % table_length将hashCode修剪为哈希表的大小来找到正确的bucket。这里有两个“陈述”，你很可能在某处读到过

如果对table_length使用2的幂，那么查找(hashCode(key) % 2^n)就像查找(hashCode(key) & (2^n -1))一样简单快捷。但是如果你为一个给定的键计算hashCode的函数不是很好，你肯定会在几个散列桶中聚集许多键。 但是，如果table_length使用质数，即使使用稍微愚蠢的hashCode函数，计算出来的hashCode也可以映射到不同的散列桶中。

这就是证明。

如果假设你的hashCode函数的结果是以下hashCode {x, 2x, 3x, 4x, 5x, 6x…}，那么所有这些都将聚集在m个桶中，其中m = table_length/GreatestCommonFactor(table_length, x)。(验证/推导这个很简单)。现在可以执行以下操作之一来避免集群

确保你不会生成太多的hashCode，这些hashCode是另一个hashCode的倍数，比如{x, 2x, 3x, 4x, 5x, 6x…}。但如果你的hashTable应该有数百万个条目，这可能有点困难。 或者通过使GreatestCommonFactor(table_length, x)等于1使m等于table_length，即使table_length与x为coprime。如果x可以是任何数字，则确保table_length是质数。

来自- http://srinvis.blogspot.com/2006/07/hash-table-lengths-and-prime-numbers.html

http://computinglife.wordpress.com/2008/11/20/why-do-hash-functions-use-prime-numbers/

解释得很清楚，还有图片。

编辑:作为一个总结，使用质数是因为当数值乘以所选质数并将它们全部相加时，获得唯一值的可能性最大。例如，给定一个字符串，将每个字母的值与质数相乘，然后将它们全部相加，就会得到它的哈希值。

一个更好的问题是，为什么是数字31?

博士tl;

Index [hash(input)%2]将导致所有可能哈希值的一半和一段值发生冲突。Index [hash(input)%prime]导致所有可能哈希值中的<2的碰撞。将除数固定为表的大小还可以确保数字不能大于表。

Primes are unique numbers. They are unique in that, the product of a prime with any other number has the best chance of being unique (not as unique as the prime itself of-course) due to the fact that a prime is used to compose it. This property is used in hashing functions. Given a string “Samuel”, you can generate a unique hash by multiply each of the constituent digits or letters with a prime number and adding them up. This is why primes are used. However using primes is an old technique. The key here to understand that as long as you can generate a sufficiently unique key you can move to other hashing techniques too. Go here for more on this topic about http://www.azillionmonkeys.com/qed/hash.html

http://computinglife.wordpress.com/2008/11/20/why-do-hash-functions-use-prime-numbers/

为了提供另一种观点，这里有一个网站:

http://www.codexon.com/posts/hash-functions-the-modulo-prime-myth

它认为你应该使用尽可能多的桶而不是四舍五入到质数桶。这似乎是个合理的可能性。直观地说，我当然可以看到桶的数量越多越好，但我无法对此进行数学论证。

Primes are used because you have good chances of obtaining a unique value for a typical hash-function which uses polynomials modulo P. Say, you use such hash-function for strings of length <= N, and you have a collision. That means that 2 different polynomials produce the same value modulo P. The difference of those polynomials is again a polynomial of the same degree N (or less). It has no more than N roots (this is here the nature of math shows itself, since this claim is only true for a polynomial over a field => prime number). So if N is much less than P, you are likely not to have a collision. After that, experiment can probably show that 37 is big enough to avoid collisions for a hash-table of strings which have length 5-10, and is small enough to use for calculations.