TOPICS
Landau's Problems
Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely:
1. TheGoldbach conjecture,
3.Legendre's conjecture that for every
there exists aprime
between
and
(Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and
4. The conjecture that there are infinitely manyprimes
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, ... (OEISA002496).
Although it is not known if there always exists aprime
between
and
, Chen (1975) has shown that a number
which is either aprime orsemiprime 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 smallestprimes between
and
for
, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEISA007491).
The first fewprimes
which areof the form
are given by 2, 5, 17, 37, 101, 197, 257, 401, ... (OEISA002496). These correspond to
, 2, 4, 6, 10, 14, 16, 20, ... (OEISA005574; Hardy and Wright 1979, p. 19).
See also
Bertrand's Postulate,Goldbach Conjecture,Good Prime,Prime Number,Twin Prime ConjectureExplore with Wolfram|Alpha
More things to try:
References
Chen, J. R. "On the Distribution of Almost Primes in an Interval."Sci. Sinica18, 611-627, 1975.Euler, L. "De numeris primis valde magnis."Novi Commentarii academiae scientiarum Petropolitanae9, 99-153, (1760) 1764. Reprinted inCommentat. arithm.1, 356-378, 1849. Reprinted inOpera 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 inAn 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. SequencesA002496/M1506,A005574/M1010, andA007491/M1389 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Landau's ProblemsCite this as:
Weisstein, Eric W. "Landau's Problems."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/LandausProblems.html