From Collatz to Black Holes

Explore the endlessly surprising landscape of iteration - where the simplest rules unleash wildly unpredictable behavior, endless fascination, and occasional startling proofs. In ’From Collatz to Black Holes: Iteration in Number Theory,’ Jason Earls takes readers on a fascinating journey through one of the most addictive corners of recreational mathematics.It begins with the legendary Collatz 3n+1 problem: start with any positive integer; if it is even, divide by 2, and if it is odd, apply 3n+1. Repeat the process and watch the sequence behave in ways that defy intuition. Will it always reach 1? No one knows, yet generations of mathematicians and enthusiasts have been captivated by the challenge.From there, Earls explores a rich collection of iterative puzzles: the reverse-and-add 196 problem that resists palindromes, digit-sum cycles, sums of squared digits, and the dramatic heights reached by juggler sequences. The book also introduces the author’s own creation - the Black Hole 14 divisor-digit iteration.Blending computation, experimentation, and accessible proofs, the book also revisits classic iterative ideas such as Newton’s square-root method and the Euclidean algorithm, alongside surprising explorations involving reversals, max digits, factorizations, and happy palindromic numbers. Historical essays bring to life figures like Fermat, Erdős, Turing, Feigenbaum, and Collatz.Simple rules. Unexpected depths. The thrill of discovery. Experiment, compute, question - and perhaps prove something new.