A practical number n is one where each number up to n can be expressed as a subset sum of the positive divisors of n. 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 φ-practical numbers n (the polynomial tn-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.
This video is part of the Number Theory Web Seminar series.
