Happy Numbers: behavior under function iteration (Spring 2019)
Let a=an…a1a0 be the decimal (base 10) expansion of a positive integer a (example: for a=437, a2=4, a1=3, a0=7). Using the decimal expansion of a, define S2(a) = an2 + … + a12 + a02. We will investigate the effect of iterating the function S2. A number a > 1 is happy if repeated application of S2 results in the number 1 (for example S2(97)=130, S2(130)=10, S2(10)=1, so 97, 130, and 10 are happy). We may also investigate behavior under related functions such as Se(a) = ane + … + a1e + a0 e for e a positive integer, or Se.b(a), which uses base-b expansion rather than base 10 expansion. All participants will learn how to use programs in the mathematical software Sage to gather data, and some may write such programs. We will look for patterns in data, ask questions, make conjectures, and try to answer or prove or find counterexamples.
People
Chad Berner (Undergrad)
Sam Fox (Undergrad)
Leslie Hogben (Faculty)
Jesse Geneson (PostDoc)
Amber Ogden (Undergrad)
Justin Stevenson (Undergrad)
Mike Ross (Grad)
Pre-requisites
Mathematical curiosity