2
votes
1answer
104 views

Probability property that the longest side of primitive Pythagorean triples is prime

If we consider the set of the first $n$ primitive Pythagorean triples, then the probability that the triple's longest side is prime is approximately $\dfrac{1}{\log_{11.475}n}$ based on Mathematica’s ...
1
vote
3answers
79 views

Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
6
votes
1answer
245 views

Probabilistic proof of existence of an integer

The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
8
votes
2answers
238 views

Probability of determinants being coprime

I have a question that is not of particular significance, but I would love to understand the underlying principles. Suppose we have two square 3x3 matrices, $M_1$ and $M_2$ with $$M_1 = ...