Question

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.

Answer 1

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.

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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!

Answer 8

thank you for your help!!!

Answer 9

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

Answer 10

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.

Answer 11

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

Answer 12

oh alright i see how that works

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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