Thanks, Aaron.  Your comment has reminded me that I have been negligent in the
computational searches conducted so far, in that I have failed to report any
information on minimum distances encountered.  I will attend to that.
<br><br>
By the way, I have reversed the definitions of <i>X</i> and <i>Y</i> above as they were the opposite
of what I have in all existing code and research notes. My apologies!
<br><br>
In terms of <i>k</i> the first few polynomials are<br><br>
$Py_1  = 4k - 1$<br>
$Px_1  = 4k + 1$
<br><br>
$Py_2  = 16k^2 - 12k + 1$<br>
$Px_2  = 16k^2 + 12k + 1$
<br><br>
$Py_3  = 64k^3 -  80k^2 + 24k - 1$<br>
$Px_3  = 64k^3 +  80k^2 + 24k + 1$
<br><br>
$Py_4  = 256k^4 - 448k^3 + 240k^2 - 40k + 1$<br>
$Px_4 = 256k^4 + 448k^3 + 240k^2 + 40k + 1$
<br><br>
If we define the distance polynomial $D_{j,i} = Py_j - Px_i$ then  $D_{2,1} = 16k^2 - 16k$ so the quadratic case is disposed of, as you say.<br><br>
We can also rule out the cubic case, and in fact all odd <i>j</i>. We have<br><br>
$D_{3,1} = 64k^3 - 80k^2 + 20k - 2$<br>
$D_{3,2} = 64k^3 - 96k^2 + 12k - 2$<br><br>
For all odd <i>j</i> we get even coefficients and $c_0 = -2$, so no $D_{2e+1,i}$ can have an integer root $k > 1$.<br><br>
For even <i>j</i> we get polys like these:<br><br>
$D_{4,1} = 56k^4 - 448k^3 + 240k^2 - 44k$<br>
$D_{4,2}= 256k^4 - 448k^3 + 224k^2 - 52k$<br>
$D_{4,3} = 256k^4 - 512k^3 + 160k^2 - 64k$<br><br>
What I'm hoping to find is some magic property for even <i>j</i> that will tell us that all $D_{2e,i}$ are either irreducible or have a single integer root $k=1$.<br><br>
Since $Y(1,j) = 3,5,7 \ldots$, all of $X(1,i) = 5, 29, 169 \ldots$ are to be found in $Y(1,j)$ so the corresponding $D_{14,2}, D_{84,3} $ etc will all have root $k=1$.<br><br>
I suspect that all other <i>D</i> are irreducible, but these isolated exceptions are a bit of a fly in the ointment!<br><br>
Oh yes, and I can tell you that a search on all pairs of sequences $Y(k,j), X(k,i)$ revealed no match for a rather staggering <i>j</i> up to 100,000. For a given depth limit <i>j < J</i>, such a search is finite, since beyond a certain <i>k</i> we find that all $Y(k,J) > X(k,J-1)$ and so we need look no further.<br><br>
It follows then that the proposition, that all $D_{j,i}$ are either irreducible or have a single integer root $k=1$ is true for all <i>j</i> < 100,000.