|It has long been known that there are infinitely many prime numbers (Euclid, 300 B.C.), but another interesting question is how close together are the prime numbers, one from the next. Or, in other words, are there, for example, more or less primes between 1 and 100 than there are between 1001 and 1100? What about 1000001 and 1000100? Does the â€œdensityâ€ of prime numbers increase or decrease as the numbers themselves get larger?
At first sight the primes seem to be distributed among the integers in rather a haphazard way. For example in the 100 numbers immediately before 10,000,000 there are 9 primes, while in the 100 numbers after there are only 2 primes. However, on a large scale, the way in which the primes are distributed is very regular. Legendre and Gauss both did extensive calculations of the density of primes. Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n).
Start End Number of Primes 1 1000 168 1001 2000 135 2001 3000 127 3001 4000 120 4001 5000 119 5001 6000 114 6001 7000 117 7001 8000 107 8001 9000 110 9001 10000 112 â€¦ â€¦ â€¦ 49001 50000 98