 ## On a New Method in Elementary Number Theory Which Leads to An Elementary Proof of the Prime Number Theorem in Proceedings of the National Academy of Sciences 35 pp. 374-384, 1949 [PRIME NUMBER THEORM]

Easton: National Academy of Sciences, 1949. 1st Edition. FIRST EDITION, bound full volume, of ERDÖS’ PRIME NUMBER THEOREM, a paper which led to Erdös' 1952 Cole Prize from the American Mathematical Society. Erdös’ elementary proof shows that prime numbers become less frequent as numbers grow large. It is ‘elementary’ in the sense that it uses no complex analysis or limiting procedures.

The study of prime numbers, specifically their distribution, has fascinated mathematicians since antiquity – they are “the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians” (Spencer, The Elementary Proof of the Prime Number Theorem, 1). The Prime Number Theorem (PNT), the theorem of Hadamard and de la Vallée Poussi, is one of the most celebrated results in analytic number theory, but both proofs are long and elaborate and employed complex analysis. The PNT “formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs.

In the year 1948 the mathematical was stunned when Paul Erdös announced that he and Atle Selberg had found a truly elementary proof of the prime number theorem. To describe the distribution of prime numbers using only the simplest properties of the logarithm function was, by some, unthinkable. Prior to Erdös and Selberg, the mathematician G.H. Hardy doubted that a less complex proof than the PNT ‘could’ ever be found.

Erdös and Selberg worked cooperatively, but ultimately published different version separately. Historians have solved the issue by noting that the discovery occurred concurrently. In 1950 both Selberg and Erdös won the prestigious Fields Medal for giving an elementary proof of the prime number theorem “with a generalization to prime numbers in an arbitrary arithmetic progression” (Fields Medal).

Paul Erdös was one of the most respected mathematicians of the 20th century; certainly he was one of the most prolific, publishing more papers than any other mathematician, including Euler. He is the subject of a television show and even a children’s book. Item #1272

CONDITION: Complete full volume, 720pp. 4to. Handsomely rebound in aged cloth to look exactly like originally issued. Bright and clean throughout. Fine condition.

Price: \$550.00

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