a^3 = amn + m + n

[Joshua Zucker]

Hi seqfans,
someone asked me to prove that a^3 = amn + m + n had no solutions in 
positive integers if a is prime.
I haven't been able to do that yet.

But, seqfan that I am, I figured I could make the sequence of all a which 
did solve this thing.

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).

And, besides those, if (a,m,n) works, then (m,?,a) also works.
For instance, since (8,30,2) works, so does (30,112,8).
And then since that works, so does (112,418,30)
And thence (418,1560,112).

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.

Can anyone help me prove that conjecture?

Meanwhile, here's the list of (a m n) that work, in case that helps.

(8 30 2)
(27 240 3)
(30 112 8)
(64 1020 4)
(112 418 30)
(125 3120 5)
(216 7770 6)
(240 2133 27)
(343 16800 7)
(418 1560 112)
(512 32760 8)
(729 59040 9)
(1000 99990 10)
(1020 16256 64)
(1331 161040 11) 
(1560 5822 418) 
(1728 248820 12)
(2133 18957 240) 
(2197 371280 13) 
(2744 537810 14)
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