# Unusual set of 3 integers

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1. Feb 14, 2016

### Terry Coates

Has anyone else spotted an unusual set of three different integers A, B, & C such that
A^n + B^n - C^n = A + B - C > 0 (n > 1 and A x B x C > 0)

I leave the reader to see if they can find this set, or to ask me what they are.

2. Feb 14, 2016

### CrazyNinja

Im assuming here that the 'n' is real and not limited to an integer. Is this allowed or are there restrictions on 'n'?

3. Feb 14, 2016

### Terry Coates

I should have said "an unusual set of four integers" as n is included. A search should find the set quite quickly since none of the integers is as high as 20.

4. Feb 14, 2016

### Samy_A

There are infinitely many solutions for n=2.

5. Feb 14, 2016

### Terry Coates

For n = 2 surely there are no solutions. 3^2 + 4^2 - 5^2 = 0 3 + 4 - 5 = 2

6. Feb 14, 2016

### Samy_A

Why is, for example, 4²+6²-7²=4+6-7=3 not a solution?

7. Feb 14, 2016

### Krylov

The product of $A, B$ and $C$ should be positive, I think.

8. Feb 14, 2016

### Terry Coates

Sorry, you are correct. I should have said n > 2. I think that then there is only one set.

9. Feb 14, 2016

### Samy_A

No, there is more than one set.

10. Feb 14, 2016

### Terry Coates

Many thanks for your speedy replies and research. My set is 16^5 + 13^5 - 17^5 = 12
I'd be pleased to see what others you have discovered.

11. Feb 14, 2016

### Samy_A

$35^3+119^3-120^3=35+119-120=34$

12. Feb 14, 2016

### Terry Coates

Thanks for that.

I actually found my set while searching for the least possible value for A^n + B^n - C^n which in the case of n = 3 the least possible = 2 (with A, B and C relatively prime) I get 64 for n = 4, 12 for n = 5, 69264 n=6, 2697354 n = 7

13. Feb 14, 2016

### PeroK

That makes it 3-0 to the Belgian.

14. Feb 14, 2016

### Staff: Mentor

Or 15-3 in Rugby counts.

15. Feb 15, 2016

### Terry Coates

Seems that these sets can only occur with power 1,2,3 and 5

16. Feb 15, 2016

### Staff: Mentor

Boldings by me.
You already ruled out 1 and 2.
???

17. Feb 15, 2016

### Terry Coates

If I add the condition that A, B and C are to be relatively prime, then my set is probably unique

18. Feb 15, 2016

### Samy_A

Not quite, these also satisfy this additional condition:
3,21,55,56
3,31,56,59
3,49,139,141
3,85,91,111
3,101,291,295

19. Feb 15, 2016

### Staff: Mentor

8-0
It's going to be a disaster ...

20. Feb 15, 2016

### nasu

It does not matter, he will keep adding conditions until his set is unique. :)