Yi came to visit UCSB today. As usual, Ryavec gave Yi all his usual favorite unsolved problems, namely his (3n+j choose n) Hankel determinant problems and Danzer’s Conjecture. For some reason, Ryavec thinks that I am good at doing arithmetic, and he kept asking me to compute binomial coefficients for him. They also discussed some combinatorial problems that I hadn’t seen before. This caused me to think of a generalization of catalan numbers: How many ways are there to get from (0,0) to (n,n) on a grid by taking right steps and up steps that cross the line y=x no more than j times (or exactly j times, for that matter)?
Another thought crossed my mind at another time during the discussion. Ryavec asked about the prime factors of a+b if we know the prime factors of a and b. Naturally, that’s a very tough (and unsolved) problem. But he wrote something like n=2^4*3^4*11^6+5^3*7^8*13^2 on the board, and it occurred to me that this number has a much greater chance of being prime than a random number that’s close to that size (i.e. 1/log(n)) since it isn’t divisible by 2, 3, 5, 7, 11, or 13.
I think I’ll do my measure theory research project on Haar measure of abelian groups. It looks interesting.
The current piece on the radio (not my current music) is a terrible insult to Vivaldi. People should really leave great works alone.