Let’s say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet: We’ll choose two positive integers at random. If they have any divisor in common (other than \(1\)) I’ll pay you a dollar, else you’ll pay me a dollar. Are you in? Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you? [Read More]
Tossing the Prime Coin
One of the problems with explaining the Riemann Hypothesis is that its fascination comes from its deep connection to prime numbers, but its definition is in terms of complex analysis which requires a fair deal of undergraduate mathematics to understand – and that is before you even got started to grasp what the heck the zeta-zeros have to do with the distribution of primes. My “cocktail party explanation” of the Riemann Hypothesis would usually be something like: “The prime numbers are as equally distributed as you could wish for. [Read More]
Are Primes Independent?
The question may sound silly, but I hope it will become apparent that it’s very reasonable to ask. What we will examine here is the probabilistic interpretation of the prime distribution. So, essentially we ask: “What’s the probability that a randomly chosen number is prime?” Those familiar with some basic probability theory know the notion of independency in this context, so the question I’m basically interested in here is if the probability to find a prime is independent of the preceding or following numbers. [Read More]