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. The Goldbach conjecture,

2. Twin prime conjecture,

3. Legendre's conjecture that for every n there exists a prime p between n^2 and (n+1)^2 (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and

4. The conjecture that there are infinitely many primes p of the form p=n^2+1 (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 p between n^2 and (n+1)^2, Chen (1975) has shown that a number P which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between n-n^theta and n where theta=23/42 (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between n^2 and (n+1)^2 for n=1, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEIS A007491).

The first few primes p which are of the form p=n^2+1 are given by 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). These correspond to n=1, 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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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."

Cite this as:

Weisstein, Eric W. "Landau's Problems." From MathWorld--A Wolfram Web Resource.

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