[2] The representations of 16 as x + y and corresponding prime divisors in X and Y are: In a 1971 paper, Paul Erdős and John Selfridge considered intervals of integers containing an element coprime to both endpoints.
They observed that earlier results of S. S. Pillai and George Szekeres implied that such an element exists for every interval of at most 16 integers; thus, no Erdős–Woods number can be less than 16.
[3] In his 1981 thesis, Alan R. Woods independently conjectured[4] that whenever k > 1, the interval [a, a + k] always includes a number coprime to both endpoints.
[6] Meanwhile, the prime-partitionable numbers had been defined by Holsztyński and Strube in 1978,[2] following which Erdős and William T. Trotter proved in 1978 that they form an infinite sequence.
Erdős and Trotter applied these results to generate pairs of directed cycles whose Cartesian product of graphs does not contain a Hamiltonian cycle, and they used a computer search to find several odd prime-partitionable numbers, including 15395 and 397197.