Find all possible positive integer(s) N, such that the decimal representation of the sum of two distinct perfect powers of N consists of non leading zeroes and contains each of the digits from 0 to 9 exactly once.

in my program I used the fact that the 2 powers would be at least 2,3 and thus n^3 can't be any larger than 9876543210 and thus need only search for n=2 to 2145.  Then for each n I search for each power that does not make n^p larger than 9876543210.  My Mathematica code is

For[n=2,n„T2145,n++,

p1=3;

While[n^p1„T9876543210,

p2=2;

While[p2<p1,

v1=n^p1;

v2=n^p2;

v3=v1+v2;

dgs=IntegerDigits[v3];

cmb=Sort[dgs];

If[cmbƒú{0,1,2,3,4,5,6,7,8,9},

Print[n,"^",p1,"= ",v1," ",n,"^",p2,"= ",v2," total: ",v3];];

p2++];

p1++];

];

and this gave the following 2 solutions

264^4=485732416 264^3=18399744 total: 4875932160

and

2016^3=819354096 2016^2=4064256 total: 8197604352

Posted by Daniel on 2009-05-06 11:51:47