Thursday, 7 March 2013

Prime number years in the 21st century

There are 14 of them: 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089 and 2099.

Although there are no known distribution patterns of prime numbers, the asymptotic distribution of primes is described by the prime number theorem:


where π(x) is the number of prime numbers less than or equal to x > 1.  Note that the prime number theorem describes only the asymptotic distribution of primes.  In other words, the bigger x is, the more accurate x/ln(x) estimates π(x) relatively.

According to the prime number theorem, the number of prime numbers between 2000 and 2100, is estimated to be:

2100/ln(2100) - 2000/ln(2000),

which is approximately 11.4, a rough estimate knowing that

π(2100) - π(2000) = 14.

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