![]() Note that as we considered the first 10, 100 and 1,000 integers, the percentage of primes went from 40% to 25% to 16.8%. Among the first 1,000 integers, there are 168 primes, so π(1,000) = 168, and so on. Similarly, π(100) = 25, since 25 of the first 100 integers are prime. This value is called π( n), where π is the “prime counting function.” For example, π(10) = 4 since there are four primes less than or equal to 10 (2, 3, 5 and 7). ![]() The prime number theorem provides a way to approximate the number of primes less than or equal to a given number n. While mathematicians never know whether a proof would merit inclusion in The Book, two strong contenders are the first, independent proofs of the prime number theorem in 1896 by Jacques Hadamard and Charles-Jean de la Vallée Poussin. One favorite is the prime number theorem - a statement that describes the distribution of prime numbers, those whose only divisors are 1 and themselves. Erdős’ mandate hints at the motives of mathematicians who continue to search for new proofs of already proved theorems. ![]() The Book, which only exists in theory, contains the most elegant proofs of the most important theorems. ![]() “You don’t have to believe in God, but you have to believe in The Book,” the Hungarian mathematician Paul Erdős once said.
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