Landau's problems are four conjectures about prime numbers posed by Edmund Landau in 1912, none of which are solved as of 2023:
Goldbach's conjecture: Every even integer (except 2) is the sum of two primes.
Twin prime conjecture: There are infinitely many primes p such that p+2 is also prime.
Legendre's conjecture: There is always a prime between any two perfect squares.
There are infinitely many primes of the form n^2+1.
Which will be the first of these conjectures to be proven or disproven?
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3.00@JosephNoonan My feeling is that it's a much harder problem. Of course the solution of any of the problems requires a breakthrough / getting past fundamental-looking barriers, but I kind of think that the n^2+1 conjecture is even further away than the others.
Note: The Goldbach's conjecture and the twin prime conjecture seem to be "morally equivalent", in that a solution of one very likely gives you a solution to the other. (Both are, more or less, about prime values of linear forms ap + b.)