Jamie Song, Mathematics
Working with Smooth Numbers
December 8, 2006
Walden Emerson Room
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Jamie Song, Mathematics Working with Smooth Numbers December 8, 2006 Walden Emerson Room |
| Questions about properties of positive integers whose prime factors are relatively small arise naturally in research of various problems in additive number theory. These natural numbers are referred to as "integers without large prime factors,'' or "smooth numbers.'' Also sometimes termed as "crumbly numbers,'' these numbers may be viewed as the opposites of prime numbers in the sense that prime numbers are naturally defined as "unbreakables''. The smooth numbers are "sifted out " numbers from the set of positive integers in the sieve of Eratosthenes in counting the number of primes. Density of smooth numbers are closely related to some of the most celebrated problems in number theory, such as the Prime Number Theorem, and the Riemann Hypothesis. It is well-known that density of smooth numbers is asymptotically similar to a continuous function. In this talk, we give an introduction to smooth numbers and their applications. |