Erdős conjecture on arithmetic progressions

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

Formally, the conjecture states that if

 \sum_{n\in A} \frac{1}{n} = \infty

then A contains arithmetic progressions of any given length. (Sets satisfying the hypothesis are called large sets.)

History[edit]

In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains arbitrarily long arithmetic progressions.[1] This was proven by Szemerédi in 1975, and is now known as Szemerédi's theorem.

In a 1976 talk titled "To the memory of my lifelong friend and collaborator Paul Turán," Paul Erdős offered a prize of US$3000 for a proof of this conjecture.[2] The problem is currently worth US$5000.[3]

Progress and related results[edit]

List of unsolved problems in mathematics
Does every large set contain arbitrarily long arithmetic progressions?

Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem. Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture.

Even the weaker claim that A must contain at least one arithmetic progression of length 3 is open. The strongest related result is due to Sanders.[4]

The converse of the conjecture is not true. For example, the set {1, 10, 11, 100, 101, 102, 1000, 1001, 1002, 1003, 10000, ...} contains arithmetric progressions of every finite length, but the sum of the reciprocals of its elements converges.

See also[edit]

References[edit]

  1. ^ Erdős, Paul; Turán, Paul (1936), "On some sequences of integers", Journal of the London Mathematical Society 11 (4): 261–264, doi:10.1112/jlms/s1-11.4.261 .
  2. ^ Problems in number theory and Combinatorics, in Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), Congress. Numer. XVIII, 35–58, Utilitas Math., Winnipeg, Man., 1977
  3. ^ p. 354, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1
  4. ^ Tom Sanders, On Roth’s theorem on progressions. Ann. of Math. (2) 174 (2011), no. 1, 619-636. [1]
  • P. Erdős: Résultats et problèmes en théorie de nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres, Fasc 2., Exp. No. 24, pp. 7,
  • P. Erdős and P.Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
  • P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., Congress Numer. XVIII(1977), 35–58.
  • P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica, 1(1981), 28. doi:10.1007/BF02579174

External links[edit]