My understanding is that it's currently unproven whether one can always place $n$ primes in $n$ intervals, as in $[1,n], [n+1,2n], \ldots, [n(n-1)+1,n^2]$. I think perhaps the first few intervals have been proven, particularly for large $n$, but the general case would also confirm a number of results so I'm guessing it remains open.
I do understand that formal proofs are important for a variety of reasons, but what I'm curious about is how the following argument, despite its informality, is not strong enough to consider as proof.
I tried placing primes in intervals as described and used a simple greedy algorithm to keep decrementing composite values until they were all prime. For $n=7$, it looks like this:
One more for comparison, with $n=23$:
The algorithm worked just fine for all naturals through $n=1500$, and it certainly gives every indication that it would continue working fine. Now, I recognize that counterintuitive results happen, particularly with large-valued combinatorial things, but I'm just not seeing a realistic way something like this could eventually fail.
With every increase in $n$, you're getting more room, greatly increasing the number of possible prime arrangements; past the first few $n$, there's no hint of any sort of dangerous-looking pseudorandom behavior that could suddenly make one of these setups unworkable.
What's the problem with accepting this sort of approach (a little more rigorous than this, obviously) as a provisional proof? I raise this issue also because I have similar feelings about other theories that seem like they should be settled, e.g. the infinitude of $x^2+1$ primes, which as far as I know, essentially nobody doubts.
But really, my primary question here is as to why absolutely overwhelming empirical evidence cannot be accepted as proof in math, particularly as we're increasingly moving into areas where some of the problems are likely to be unprovable anyway. I know the cautionary tales about large numbers and strong trends being overturned, but I can't believe that applies to something as clear as, say, the validity of Legendre's conjecture.
"why absolutely overwhelming empirical evidence cannot be accepted as proof in math"
Because overwhelming empirical evidence of a statement just proves that counterexamples are rare, not that they don't exist. There are so many examples of this happening in mathematics that I won't list them here.
I guess the point is that you have to decide what the point of mathematics is. If you want to build a bridge, or use cryptography, then you are probably able to use highly likely, but not completely proved results. Cryptography uses P$\neq $NP all the time. If it turns out that our intuition is wrong, then much of our information security is completely destroyed.
But if you are trying to decide what is true, rather than what is useful, you are a pure mathematician. An applied mathematician, a physicist, an engineer, would accept that Legendre's conjecture is true, certainly for any value they are going to use it. They just don't need to worry if it fails for $n=10^{10^{100}}$.
That's a perfectly valid viewpoint, but it's just not the definition of pure mathematics. Statements with overwhelming evidence but no formal proof are called conjectures.
Edit: are you happy with the statement 'there are no more Fermat primes'? There's overwhelming empirical evidence that there are no more, but I would feel very uncomfortable declaring that it was definitely true.