Fortunate number Fortunate number A Fortunate number, named after Reo Fortune, is the smallest integer "m" > 1 such that, for a given positive integer "n", "p"# + "m" is a prime number, where the primorial "p"# is the product of the first "n" prime numbers. For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to
Prime number theorem Dusart proved: The proof by de la Vallée Poussin implies the following. For every , there is an such that for all , As a consequence of the prime number theorem, one gets an asymptotic expression for the th prime number, denoted by : A better approximation is Again considering the th prime number , this gives an estimate of ; the first 5 digits match and relative error is about 0.00005%. Rosser's theorem states that This can be improved by the following pair of bounds: The table compares exact values of to the two approximations and . The last