The modern manifestation is mostly the intellectual product of Konrad Zuse, who wrote "digital physics" in 1969.

> https://en.wikipedia.org/wiki/Digital_physics

Wolfram, Tegmark, Bostrom, etc. are mostly downstream of Zuse.

   >   Fermat Reincarnation.

That’s fine for GIMPS, which only searches for Mersenne primes, but doesn’t work in general.

https://en.wikipedia.org/wiki/Primality_test#Fast_determinis... mentions several tests that do not require factorization, though.

A and B are statements within a given set of axioms whose truth value is knowable within those axioms. A could be something like "Some number k that we pick is prime", and B could be something like "k is even"

When you see in math people saying "some statement is true iff some other statement is true", that "iff" stands for "if and only if", which really just means two things hold:

1. Starting with statement A, we can prove statement B.

2. Starting with statement B, we can prove statement A.

In math shorthand, we'd write this as

1. A implies B, or just A => B

2. B => A

You need to prove both directions for you to be able to say "A iff B"

Let's try it with our example statements. Does it hold? Your intuition should be saying "absolutely not", and let's see why:

1. k prime implies k is either 2, or odd. So the statement A => B only holds when k is 2. We choose k, so this could be true in a trivial case, but does not hold in the generic case

2. k even implies k is divisible by 2, so again, the statement "k even => k prime" only holds for one trivial case and not in the generic one.

Now for the original comment. I was pointing out that just because you have some proof of A iff B, does not mean there couldn't be another, completely separate statement C, for which you can prove A iff C. These relationships have equivalence, but are nonetheless not the same (outside of a categorical sense of sameness).

Some of the most compelling math of the 20th century was showing the sameness of many different fields by finding new iff relationships.