The Deranged Mathematician: Ulam Sequences

One way to think about this post is that it is the culmination of my series on Surviving Proofs. But it is also a standalone article about how a team of undergraduates and I solved an interesting problem in additive number theory. It strikes me that it might be instructive for some to see our thought process traced, showing how things are actually proved in mathematics in practice. The problem itself involves Ulam sequences, which are defined as follows: an Ulam sequence starts with two integers 0< a < b , and then every subsequent term is the next smallest integer that is the sum of two previous terms in exactly one way. Despite the comparatively simple combinatorial definition, even very basic questions (e.g., how fast does such a sequence grow?) are still open 60 years after Ulam originally posed them. But we have discovered some truly startling patterns in such sequences in the intervening years. Read the full post (for free) on Substack: Ulam Sequences