Edmund Landau's Problems Landau's problems are four conjectures about prime numbers which were unsolved at the time Edmund Landau presented them at the International Congress of Mathematicians in 1912.
They include:


*

*Goldbach's conjecture

*Twin prime conjecture

*Legendre's conjecture

*Conjecture that there are infinitely many primes of the form p = n^2 + 1.


It is a hundred years later and I think all four problems are still unsolved.
Is this statement true? Does anyone know if there is any recent progress on any of these?
Thanks for any additional insights or pointers.
-A
 A: The answer to the first two questions can be found in (for example) Wikipedia by searching under "twin primes" and "Goldbach," respectively. According to the articles there, the first two problems remain unsolved. OEIS sequence A002496 lists the elements of the sequence $\{n^2 +1 | n^2+1 = p\}$ with a comment (in 2001) that the sequence is conjectured but not proven to be infinite. 
Specialists in questions 1, 2, and 4 might address the question of "recent progress" with respect to these. The questions themselves pre-date Landau's discussion. According to Pintz, below, the "twin prime" conjecture may date to the time of Euclid. There are hundreds of papers dealing with these questions. They are easy to understand but extremely difficult to prove.
Legendre's conjecture remains unproven. Papers by Ingham (1937) and Cheng (2010) prove there is a prime between $n^3$ and $(n+1)^3,$ and a 1975 paper by Chen proves there is a prime or a near-prime on square intervals. There are dozens of other papers with nice results about primes on (sort of) small intervals but square intervals are currently out of reach.  
Questions 3 and 4 are not unrelated, since a sequence of the type in question 4 could not impinge on a (hypothetical) countable sequence of prime-free square intervals.    
While I have only started reading it, there is an authoritative review of Landau's problems in: Janos Pintz, Landau's Problems on Primes, Journal de theorie des nombres de Bordeaux, 2009.
[My computer is giving me a "math processing error" so I am leaving the answer here. Edits invited. Will add to this if a reboot solves.] 
