Can you solve this variant of Dragan K's equation?

Question

Find distinct positive integers a,b,c such that a^3+b^3 = c^3+11c.

Dragan K · Accepted Answer

I can find only one triple:

755^3 + 6913^3 = 6916^3 + 11*6916

Probably there are more solutions, but they are huge.

*************************
I found it! :)

1135^3 + 1157^3 = 1444^3 + 11*1444

String · Answer

I know this is too late, as I haven't solved the problem but only have one minor observation that might start off some other bright head:

cÂ³ + 11c = (c-1)Â·cÂ·(c+1) + 12c

and since c-1, c and c+1 are three consecutive integers at least one of them is divisible by 2 and at least one is divisible by 3. Since 12c is always divisible by 6 as well the entire expression is divisible by 6. Hence if cÂ³+11c = aÂ³+bÂ³ is divisible by 6 then aâ¡-b mod 6.

If c-1 or c+1 is divisible by 12 then the expression is divisible by 12c obviously.

Can you solve this variant of Dragan K's equation?

Trending Questions