## jennwatso15 2 years ago Prove that there are no positive perfect cubes less than 1000 that are the sum of the cubes of two positive integers? I know all the cubes greater than zero and less that 1000.

1. gianarpt

The perfect cubes less than 1000 are: 1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 6^3 = 216 7^3 = 343 8^3 = 512 9^3 = 729 What we want to prove is that none of these is the sum of any other two. This involves a simple (but tedious) checking of the list. Clearly a number could only be the sum of two numbers unless both were strictly smaller than it, so we only need to consider smaller terms. Thus 1 can't be the sum of any other terms. 2^3 = 8 could only be 1 + 1, but this is 2, not 8. 3^3 = 27 could be 1 + 1, 1 + 8 or 8 + 8, but all of these are too small 4^3 = 64 could be 1 + 1, 1 + 8, 1 + 27, 8 + 8, 8 + 27, 27 + 27... but all of these are too small Continuing in this way you can verify the proposition. There is a general proof that a^3 + b^3 = c^3 holds for no positive integers (and no, I don't mean the full proof of Fermat's last theorem) but it's too involved to post here.

2. jennwatso15

alright that's a little bit clearer than what the book gave. thank you

3. gianarpt

sorry if its too much, i was just trying to help you understand it better! hope it helped, and you're welcome :)

4. jim_thompson5910

to sum up what gianarpt said there are no integer or whole number solutions a, b, c that satisfy the equation a^3 + b^3 = c^3 and this is based off of fermat's last theorem http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

5. jim_thompson5910

so there's no way to find two integers, say x and y, that make the equation x^3 + y^3 = 8^3 true for instance

6. jennwatso15

it kinda did. it would help if we went over this in class yet but we have not so i am hoping he doesn't take up the homework. what's not making sense is why your doing the one plus one and stuff. because the example is different i'm not proving that there isn't any

7. gianarpt

Well I hope you get better at this and figure it out and pass your quiz/homework :) But sadly I have to get off but thanks for the medal also Jim!

8. jennwatso15

9. jim_thompson5910

yw gianarpt the 1+1, and all that is adding up the cubes 1, 8, 27, etc to try to get another cube 1+1 (sum of 2 cubes) = 2 but 2 is NOT a perfect cube

10. jennwatso15

and sorry if i'm becoming difficult I don't know much about proofs at all I was thrown in this class in college without taking calc 1 first so sorry if i'm slow at this.

11. jim_thompson5910

1+8 = 9, but 9 isn't a perfect cube you'd be stuck looking forever trying to find a perfect cube that breaks down into x^3 + y^3 because there aren't any

12. jennwatso15

oh alright i see how that works

13. jim_thompson5910

the justification I would use is fermat's last theorem the proof is very hard to understand...it took about 400 years (or so) for someone to figure it out...and it was probably the most famous and well known problems in math

14. jennwatso15

i'm debating on if I should write all of these out in the homework it now makes sense on what she first responded

15. jim_thompson5910

then write whatever makes the most sense to you, that's what I would do

16. jennwatso15

alright! well thank you! the question makes alot of sense now