Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely:
1. The Goldbach conjecture,
3. Legendre's conjecture that for every 
there exists a prime 
between 
and 
(Hardy and Wright 1979, p. 415; Ribenboim 1996,
pp. 397-398), and
4. The conjecture that there are infinitely many primes 
of the form 
(Euler 1760; Mirsky 1949; Hardy
and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such
primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496).
Although it is not known if there always exists a prime 
between 
and 
, Chen (1975) has shown that a number 
which is either a prime or
semiprime does always satisfy this inequality. Moreover,
there is always a prime between 
and 
where 
(Iwaniec and Pintz 1984; Hardy and Wright 1979,
p. 415). The smallest primes between 
and 
for 
, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEIS
A007491).
The first few primes 
which are of the form 
are given by 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS
A002496). These correspond to 
, 2, 4, 6, 10, 14, 16, 20, ... (OEIS A005574;
Hardy and Wright 1979, p. 19).
See also
Bertrand's Postulate, Goldbach Conjecture, Good Prime, Prime Number, Twin Prime Conjecture
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References
Chen, J. R. "On the Distribution of Almost Primes in an Interval." Sci. Sinica 18, 611-627, 1975.Euler, L. "De numeris primis valde magnis." Novi Commentarii academiae scientiarum Petropolitanae 9, 99-153, (1760) 1764. Reprinted in Commentat. arithm. 1, 356-378, 1849. Reprinted in Opera Omnia: Series 1, Volume 3, pp. 1-45.Goldman, J. R. The Queen of Mathematics: An Historically Motivated Guide to Number Theory. Wellesley, MA: A K Peters, p. 22, 1998.Hardy, G. H. and Wright, W. M. "Unsolved Problems Concerning Primes." §2.8 and Appendix §3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 19 and 415-416, 1979.Iwaniec, H. and Pintz, J. "Primes in Short Intervals." Monatsh. f. Math. 98, 115-143, 1984.Ogilvy, C. S. Tomorrow's Math: Unsolved Problems for the Amateur, 2nd ed. Oxford, England: Oxford University Press, p. 116, 1972.Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 132-134 and 206-208, 1996.Sloane, N. J. A. Sequences A002496/M1506, A005574/M1010, and A007491/M1389 in "The On-Line Encyclopedia of Integer Sequences."
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Cite this as:
Weisstein, Eric W. "Landau's Problems." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LandausProblems.html