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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:

  1. Goldbach's conjecture
  2. Twin prime conjecture
  3. Legendre's conjecture
  4. 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

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    $\begingroup$ According to Wiki, the first two are unsolved. Per OEIS 002496 the last is unsolved. And Legendre's conjecture is still a conjecture. $\endgroup$
    – daniel
    Nov 26, 2012 at 23:53
  • $\begingroup$ I noticed this (arxiv.org/abs/1211.6046) was uploaded to arxiv very recently. Not being an expert I have absolutely no idea as to whether it is correct or not, however it may at least contain references to other recent progress on Legendre's conjecture. $\endgroup$
    – Alex J Best
    Nov 27, 2012 at 23:03
  • $\begingroup$ He cites one paper, by Rosser-Schoenfeld, from which I think he uses a bound for Chebyshev's first function. $\endgroup$
    – daniel
    Nov 27, 2012 at 23:43
  • $\begingroup$ Yeah I noticed that after I posted, perhaps not so useful after all! Still it seemed like too much of a coincidence not to post after it came in the arxiv email this morning. $\endgroup$
    – Alex J Best
    Nov 27, 2012 at 23:56
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    $\begingroup$ Very useful if it's right. $\endgroup$
    – daniel
    Nov 28, 2012 at 0:01

1 Answer 1

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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.]

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  • $\begingroup$ "They are easy to understand but extremely difficult." What does that mean?? $\endgroup$
    – TonyK
    Nov 27, 2012 at 1:30
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    $\begingroup$ I think he means they are very easy to understand (meaning), but so far, have proven too difficult to figure out a proof. $\endgroup$
    – Amzoti
    Nov 27, 2012 at 1:31

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