If the sum of the reciprocals of a set of positive integers diverges, then the set contain arbitrarily long finite arithmetic progressions.
~Formally~
If \(A\) is a large set in the sense that $$ \sum_{n \in A}{\frac{1}{n}}=\infty $$ then \(A\) contains arithmetic progressions \( \{a, a+c, \ldots , a+kc\}\subset A \) for arbitrarily large \(k\).
| Author | Date Published | Prize | Paid By |
| Paul Erdős, Pál Turán [1] | 1936 | $5,000 | Erdős |
[1] P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
Erdős–Turán conjecture; arithmetic progressions