Carl Pomerance: Practical numbers (NTWS 027)
Channel: Number Theory Web Seminar
Date: 2023-05-28

Abstract: A practical number n is one where each number up to n can be expressed as a subset sum of n's positive divisors. It seems that Fibonacci was interested in them since they have the property that all fractions m/n with m strictly less than n can be written as a sum of distinct unit fractions with denominators dividing n.  With similar considerations in mind, Srinivasan in 1948 coined the term "practical". There has been quite a lot of effort to study their distribution, effort which has gone hand in hand with the development of the anatomy of integers.  After work of Tenenbaum, Saias, and Weingartner, we now know the "Practical Number Theorem": the number of practical numbers up to x is asymptotically cx/log x, where c= 1.33607....  In this talk I'll discuss some recent developments, including work of Thompson who considered the allied concept of phi-practical numbers n (the polynomial t^n-1 has divisors over the integers of every degree up to n) and the proof (joint with Weingartner) of a conjecture of Margenstern that each large odd number can be expressed as a sum of a prime and a practical number.

Link to slides: https://drive.google.com/file/d/15ves8py2sdraRYJJ7T0PJXh0JsLpSe8l

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, August 13, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 14, 2020 (1am AEST, 3am NZST)