The Erdős-Woods conjecture asserts that there is a fixed positive integer 
such that two positive integers 
and 
must be equal whenever 
and 
have exactly the same prime
factors for every 
(Erdős 1963, Shorey and Tijdeman 2016). Thus
a sufficiently long block of consecutive integers would
be determined by its ordered pattern of prime factors.
The conjecture remains open. The stronger three-term question recorded as problem 850 asks whether 
suffices (Bloom 2026). This is distinct from the endpoint
divisibility property defining an Erdős-Woods
number.
