A pair of sequences of positive integers a, a+1,..., a+k and
b, b+1,...,b+k is called Erdös-Woods pair if for each
i less or equal to k, the integers a+i and b+i have the same prime
factors; in this case we say also that the pair have a depth
k.
Our talk is devoted to the following statement knonwn as Erdös-Woods Conjecture: for some positive integer k, there is no any Erdös-Woods pair of depth k. Woods noticed that this purely number-theoretic conjecture is closely related to some questions of the formal arithmetics and its fragments.