Erdős Problems

Tags Prizes

OPEN

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?

#850: [Er63,Problem 60][Er80f][Er96b]

number theory, primes

This is sometimes known as the Erdős-Woods conjecture.

For just $x,y$ and $x+1,y+1$ one can take \[x=2(2^r-1)\] and \[y = x(x+2).\] Erdős also asked whether there are any other examples. Makowski [Ma68] observed that $x=75$ and $y=1215$ is another example, since \[75 = 3\cdot 5^2 \textrm{ and }1215 = 3^5\cdot 5\] while \[76 = 2^2\cdot 19\textrm{ and }1216 = 2^6\cdot 19.\] (This example was also recently found independently by Matthew Bolan.) No other examples are known. This sequence is listed as A343101 at the OEIS.

Shorey and Tijdeman [ShTi16] have shown that, assuming a strong form of the $abc$ conjecture due to Baker, then the answer to the original problem is no.