ParthKohli 2 years ago Solving the diophantine equation:\[a^3 + b^3 + c^3 = 100a + 10b + c \]

1. perl

hmmm

2. ParthKohli

Can you help me with that?

3. perl

find all integer solutions

4. perl

looks like we can factor something here

5. perl

did you try to expand (a+b+c)^3

6. ParthKohli

Whoo.

7. ParthKohli

\[a^3+3 a^2 b+3 a^2 c+3 a b^2+6 a b c+3 a c^2+b^3+3 b^2 c+3 b c^2+c^3\]

8. ParthKohli

So do we have to add \(3 a^2 b+3 a^2 c+3 a b^2+6 a b c+3 a c^2+3 b^2 c+3 b c^2\) to both sides?

9. ParthKohli

\[(a + b + c)^3 = 3 a^2 b+3 a^2 c+3 a b^2+6 a b c+3 a c^2+3 b^2 c+3 b c^2 + 100a + 10b + c \]

10. perl

oh look at that

11. perl

one second

12. perl

(a+b+c)^3 = a^3+3 a^2 b+3 a^2 c+3 a b^2+6 a b c+3 a c^2+b^3+3 b^2 c+3 b c^2+c^3

13. ParthKohli

How would you solve it now?

14. perl

not sure

15. perl

where did you get this question, is it solvable?

16. perl

well the simple approach is , equate terms

17. ParthKohli

Yes, a solution is \((1,5,3)\)

18. perl

|dw:1360413980599:dw|

19. perl

so if a^2 = 100 , b^2 = 10 , and c^2 = 1 , you have a solution

20. ParthKohli

I didn't mention that \((a,b,c)\) all must be single-digit numbers.

21. perl

ok, then you can go through all the cases , 0-9

22. ParthKohli

I want a purely mathematical solution :-|

23. ParthKohli

Not by guesses...