Question

Complete the following little proof showing that y^3 = x^2 + 2 has no solutions over integers other than x=5, y=3

Answer 1

\(\begin{align}y^3 &= x^2+2\\~\\
&=x^2 - (i\sqrt{2})^2\\~\\
&=(x+i\sqrt{2})(x-i\sqrt{2})
\end{align}\)


Each of the right hand side factors must be a perfect cube since the left hand side is a perfect cube. In particular, we take :
\[\begin{align}x+i\sqrt{2} &= (a+ib\sqrt{2})^3\\~\\
&=\cdots
\end{align}
\]

Answer 2

x=3 y=2 is not a solution.. ?

Answer 3

dang :( doing stuff i don't know ;-;

Answer 4

sorry il update, it is
x=5, y=3

Answer 5

you do know these stuff @ILovePuppiesLol
i have just phrased the question in a way that you wont understand easily.. ;)

Answer 6

can you clear it up for me so i don't sit here like a pregnant koala trying to swim, please :)

Answer 7

In simple words, here is the problem :
```
Find all the whole numbers that satisfy the equation y^3 = x^2 + 2
```

Answer 8

ohhh

Answer 9

plugin x = 5, y = 3 and convince yourself that it is indeed a solution

Answer 10

yes, it is, wait what are we trying to figure out again, as many solutions as possible?

Answer 11

I challenge you to find just one more solution

Answer 12

will substitution work, or will it take me about a year

Answer 13

x=-5 xD

Answer 14

y=3

Answer 15

Very very clever puppy!

Answer 16

:O I did it

Answer 17

Here is your medal for being alert

Answer 18

yay!

Answer 19

what do we do now?

Answer 20

Try finding another solution

Answer 21

another one? UGHHH

Answer 22

im stuck

Answer 23

y^3-x^2 = 2 is basically what your equation says right?

Answer 24

Yes, im deleting the graph it is making the thread slow..

Answer 25

I did it! Followed through and got \(x=\pm 5, y=3\) as the solution. The way they go complex in this proof is pretty new to me, it's interesting! :)

Answer 26

Awesome! It must have felt amazing finishing off the proof :D

Answer 27

Haha, I'm afraid the cool bit was done by you, it was great to learn about the method but the proof was relatively trivial after that ;P

Answer 28

solution please :|
is there any other way to do it?

Answer 29

without complex numbers...

Answer 30

solve \[y^3-x^2 = 2\] @baru

Answer 31

I find you can just graph it, that easily shows you there can be at most two real solutions. lol

Answer 32

For every value of x there is a corresponding value of y
So there are infinitely many real solutions