prime number theorem

prime number theorem [¦prīm ¦nəm·bər ‚thir·əm]

(mathematics)

The theorem that the limit of the quantity [π(x)] (lnx)/xasxapproaches infinity is 1, where π(x) is the number of prime numbers not greater thanxand lnxis the natural logarithm ofx.

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(mathematics)

prime number theorem - The number of prime numbers less than x is about x/log(x). Here "is about" means that the ratio of the two things tends to 1 as x tends to infinity. This was first conjectured by Gauss in the early 19th century, and was proved (independently) by Hadamard and de la Vall'ee Poussin in 1896. Their proofs relied on complex analysis, but Erdös and Selberg later found an "elementary" proof.