Hi seqfans,<br>
someone asked me to prove that a^3 = amn + m + n had no solutions in positive integers if a is prime.<br>
I haven't been able to do that yet.<br>
<br>
But, seqfan that I am, I figured I could make the sequence of all a which did solve this thing.<br>
<br>
It turns out that a = x^3 works, with m = x^5 - x and n = x (of course
m and n are interchangeable; let's just call m the larger of the two).<br>
<br>
And, besides those, if (a,m,n) works, then (m,?,a) also works.<br>
For instance, since (8,30,2) works, so does (30,112,8).<br>
And then since that works, so does (112,418,30)<br>
And thence (418,1560,112).<br>
<br>
It appears that every solution -- at least up to a = 3000, which is as
far as I checked -- has one of these forms: either a = x^3, or the
solution is generated from a smaller one.<br>
<br>
Can anyone help me prove that conjecture?<br>
<br>
Meanwhile, here's the list of (a m n) that work, in case that helps.<br>
<br>
(8 30 2)<br>
(27 240 3)<br>
(30 112 8)<br>
(64 1020 4)<br>
(112 418 30)<br>
(125 3120 5)<br>
(216 7770 6)<br>
(240 2133 27)<br>
(343 16800 7)<br>
(418 1560 112)<br>
(512 32760 8)<br>
(729 59040 9)<br>
(1000 99990 10)<br>
(1020 16256 64)<br>
(1331 161040 11) <br>
(1560 5822 418) <br>
(1728 248820 12)<br>
(2133 18957 240) <br>
(2197 371280 13) <br>
(2744 537810 14)<br>
<br>