Definition: "easy" number theory learnt in primary school, notably the operations of addition, subtraction, multiplication and division.

Can be calculated efficiently with the Extended Euclidean algorithm.

The beauty of this algorithm is that because exponentiation grows really fast, there is no hope that we can ever learn all the digits of an exponential, as there is simply not enough time or memory for that. Therefore, a natural sub-question is if we can know some part of that number, and knowing the smallest digits is the most natural version of that question.

math.stackexchange.com/questions/2382011/computational-complexity-of-modular-exponentiation-from-rosens-discrete-mathem mentions:can be calculated in:Remember that $log(m)$ and $log(n)$ are the lengths in bits of $m$ and $n$, so in terms of the length in bits $M$ and $N$ we'd get:

There are infinitely many prime k-tuples for every admissible tuple.

Generalization of the Twin prime conjecture.

As of 2023, there was no specific admissible tuple for which it had been proven that there infinite of, only bounds of type:But these do not specify which specific tuple, e.g. Yitang Zhang's theorem.

There are infinitely many primes with a neighbor not further apart than 70 million. This was the first such finite bound to be proven, and therefore a major breakthrough.

This implies that for at least one value (or more) below 70 million there are infinitely many repetitions, but we don't know which e.g. we could have infinitely many:or infinitely many:or infinitely many:or infinitely many:but we don't know which of those.

The Prime k-tuple conjecture conjectures that it is all of them.

Also, if 70 million could be reduced down to 2, we would have a proof of the Twin prime conjecture, but this method would only work for (k, k + 2).

Let's show them how it's done with primes + awk. Edit. They have a -d option which also shows gaps!!! Too strong:gives us the list of all twin primes up to 100:Tested on Ubuntu 22.10.