# Econstudentlog

“In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a rough description of how the primes are distributed.

Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. In other words, the average gap between prime numbers near N is roughly ln(N).”

‘Roughly’ because we’re dealing with asymptotics. For n=10, the true likelihood is 40%, the approximative result is 43,43 percent [10/ln(10)], for n=100 the true likelihood is 20%, the approximative result is 21,71% [100/ln(100)]. There’s more stuff along the same lines here. x/ln(x) is not the best estimator of pi(x), but it works.

The right hand side of the article has a lot of great links to articles belonging to the article series on ‘Military of ancient Rome’. If you find this subject interesting, there’s probably a lot of stuff waiting for you there.

4) Hayashi limit (the maximum radius of a star for a given mass).

5) Placoderms. They died out 360 million years ago, yet we still know that they existed and even a bit more than that. “A 380-million-year-old fossil of one species represents the oldest-known example of live birth.” The fact that we even know something like that – how cool is that? Very cool! Here’s a picture: