Im completely stuck

[tired scarab]

can you factor x^2 - y^2

[proud trellis]

Oh ok so $(x+y)(x-y) = 1, (x+y) = (x-y)$
$2y = 0, y = 0$
$x^2 = 1$

[modern shore]

Kardiiacc

[proud trellis]

am i on the right line?

[tired scarab]

how are you going from (x+y)(x-y) = 1 to (x + y) = (x-y)?

you are certainly going to reason about x and y from the first step that you did

but the right line here is just to think about what it means for two integers to multiply to 1

[proud trellis]

Yo my apologies i literally made an error when calculating this ima do it again soz

[tired scarab]

it's okay!

but you are done with the symbolic manipulation once you factor in the first step

you just need to realize that it's not possible for (x+y)(x-y) to ever be 1 (why is this true?)

(remember that x and y are positive integers)

[proud trellis]

Bc nothing multiplies together to get 1 except 1 and 1

[tired scarab]

exactly (or -1 and -1, but you will never have that case, since x and y are positive)

[tired scarab]

so why can't both factors be 1

what if i told you that both factors were 1 what would you say to me

[proud trellis]

x would have to be negative as its both -y and +y to get 1

or 0*

Oh wait i think i got that wrong

[tired scarab]

there is a clearer reason

think about what we supposed for the sake of contradition - that x and y were both positive integers

[proud trellis]

So (x+y) = 1 and (x-y) = 1 and by sake of contradiction, there is atleast on positive integer that has the solution x^2-y^2 = 1. Use simulataneous equation to solve for y. 2y = 0, y = 0. Plug in y and x = 1. but there is atleast one positive integer tho

[proud trellis]

Im still a bit stuck

[tired scarab]

well what would it mean for x + y = 1

specifically

we can get a contradiction right here

[proud trellis]

Ohhh there is no positive integer that can add to get 1

[tired scarab]

yep

[proud trellis]

Lolll i didnt even realise

[tired scarab]

lol i remember doing this problem for the first time and it was really confusing to me

and then i saw it and I was like omg

okay so you should write out the full contradiction proof (either on your paper or here) and then you're done

are you comfortable with writing a contradiction proof?