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?
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. Matthew Bolan has observed that $x=75$ and $y=1215$ is another example, since
\[75 = 3\cdot 5^2 \textrm{ and }1215 = 3^5\cdot 5\]
while
\[76 = 2^2\cdot 19\textrm{ and }1216 = 2^6\cdot 19.\]
No other examples are known. This sequence is listed as
A343101 at the OEIS.
See also [677].
