Does Erdos' Unresolved Conjecture Involve n-term Arithmetic Progressions?

[Dragonfall]

MATHWORLD: "Erdos offered a prize for a proof of the proposition that 'If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic progressions.' This conjecture is still open (unsolved), even for 3-term arithmetic progressions. "

What's an n-term arithmetic progression?

 

[shmoe]

Arithmetic progression-consecutive terms differ by a constant amount. x,x+d,x+2*d is a 3 term arithmetic progression, x, x+d, ..., x+(n-1)*d is an n-term arithmetic progression.

Compare Erdos conjecture with Szemeredi's theorem on arithmetic progressions, which makes a stronger assumption about your subset of the integers. Also Green and Tao's result on primes containing arbitrarily long arithmetic progressions as a special case of Erdos conjecture (sum of the reciprocals of the primes diverges)

 

[tacman]

An n-term arithmetic progression is a sequence of numbers where each term differs from the previous one by a constant value, known as the common difference. For example, 2, 5, 8, 11, 14 is a 5-term arithmetic progression with a common difference of 3. In general, an n-term arithmetic progression can be represented as a, a+d, a+2d, ..., a+(n-1)d, where a is the first term and d is the common difference.