lorem ipsum

Reflexively admitting to a dislike of mathematics is equivalent to declaring yourself an atheist, for God is a mathematician.
keywords: belief, faith, ability

lorem ipsum

The sacred and the fixed-but-arbitrary are two sides of the same coin, and making progress depends on adopting a focus on one or another. (Instances of a tomb of an unknown soldier combine these two foci, like an ellipse, and hence have a strong grip on the popular imagination.) People like C. S. Lewis focus on the sacred. People like C. F. Gauss focus on the fixed-but-arbitrary.
(cf: “A man must adhere to rise.”)
keywords: productivity, longing

Collatz and company

Conjecture: The Collatz conjecture is just a disguised way of asking for a formula for the n-th prime, and will never be solved. You could make up other disguised ways of asking or a formula for the n-th prime. For example: let w be a positive integer. Then, let x(1) = w, and for each positive integer n > 1, if x(n – 1) = 1, stop. Otherwise, if x(n – 1) is square-free, let x(n) be the sum of the exponents in the prime factorization of x(n – 1), and if x(n – 1) is not square-free, let x(n) be 1 + x(n – 1). Then: Does this process stop for each and every starting positive integer w?
keywords: Mathematics, Number Theory