Question

how can i find the ordered pairs of x^3 - y^3 = 721?

Answer 1

what are ordered pairs

Answer 2

u wanan know the values of x and y that work for this?

Answer 3

yes, I want to know the work as to how i can get to them

Answer 4

i see

Answer 5

i dunno lol weird question, thers a bunch of ways for finding some ordered pairs

Answer 6

cant u just solve the equation for where y = 1 , y = 2 , y=3

Answer 7

they have to be integers

Answer 8

well that will give u values of x too

Answer 9

There are only 4 (x,y) pairs

Answer 10

i dunno agents giving u an answer tho

Answer 11

Hmm, you could start by using a difference of two cubes\[\Large x^3 - y^3 = (x-y)(x^2+xy+y^2) \]
Since you want integers, you could find the factors of 721, since \[\Large (x-y)(x^2+xy+y^2) = 721\]
means some integer times some integer equals 721.

Factors of 721: 1, 7, 103, and 721 (I just googled them)

Answer 12

So we have \[\Large 1 \times 721 = 721\]\[\Large 7 \times 103 = 721\]
\[\Large (x-y)(x^2+xy+y^2) = 721\]
Let's first do 1*721... this gives \[\Large x- y = 1\]and \[\Large x^2+xy+y^2 = 721\]

Or the other way around... \[\Large x- y = 721\]and \[\Large x^2+xy+y^2 = 1\]
You can solve these since they're simultaneuous equations... oh and they also could both be negative... (-1)(-721)= 721...this gives you a couple more equations to solve (note they may not all have valid solutions).

Answer 13

Use substitution to solve those equations, and these ones.

Now do it again for 7*103=721... giving \[\Large x- y = 7\]and \[\Large x^2+xy+y^2 = 103\]

or
\[\Large x- y = 103\]and \[\Large x^2+xy+y^2 = 7\]

And again, (-7)(-103) = 721 so you may have to repeat them with negatives to find all solutions.

Answer 14

oh i see u want only integer solutions

Answer 15

Thank you!

Answer 16

You're welcome. It'll be kinda tedious solving all those equations (there's really 8 total, if you check the negatives too, which you might need to). But it should give you your integer solutions.

Answer 17

ok, thanks

Answer 18

Oh and I'm sure some of those might give non-integer solutions, which you can discard, or they might give you extraneous solutions.