Question

Fermat's Last Theorem 3 Proof Help

Answer 1

Here is the proof given

Answer 2

Answer the following:
1. From Part 2, prove that gcd(x0, y0) = 1.
2. In Part 3 the author writes “If z0 is even then x0 and y0 are odd.” What justifies this statement?
3. In Part 3, find expressions for x0 and y0 in terms of p and q.
4. From Part 3, use gcd(x0, y0) = 1 to prove that gcd(p, q) = 1.
5. From Part 3, prove that p not≡ q (mod 2).
6. From Part 3, prove that z30 = 2p(p2 + 3q2).
7. From Part 4, prove that when z0 is odd, there exist p and q so that (a) gcd(p, q) = 1 (b) p not≡ q (mod 2) (c) 2p(p2 + 3q2) is a cube (Hint: begin with “Since z0 is odd, exactly one of x0 or y0 must be even. Without loss of generality, suppose x0 is even.”)
8. In Part 7, what mathematical statements should appear in the first two blanks and what proposition should be cited in the third blank?
9. In Part 7, verify that 2p = (2c)(c − 3d)(c + 3d).
10. In Part 8, verify that x^3 + y^3= z^3.
11. In Part 8, justify that z1 < z0.

Answer 3

any ideas would be helpful

Answer 4

i am not sure exactly what you are asking, and this is rather long. but part two says you can assume gcd(x,y)=1, and that is because if it is not 1, that is if they have a common factor, then so does
\[x^3, y^3\] and therefore
\[z^3\] has a common factor with the first two,. so you can reduce and find smaller ones

Answer 5

in part 3
if z^3 is even then x^3, y^3 are odd because if z^3 is even and say x^3 is even then y^3 would also have to be even (since even number plus odd number is odd) and therefore all three would be even contradicting the fact that they are relatively prime

Answer 6

ok that makes sense thx

Answer 7

\[x_0=2p-y_0\]

Answer 8

oh sorry, in terms of p and q got it

Answer 9

\[ 2p = x_0 + y_0\]
\[ 2q = x_0 − y_0\]
\[2x_0=2p+2q\]
\[x_0=p+q\]

Answer 10

ya for that one i thought it was x0 = p+q and y0 = p-q
not sure if that's correct though

Answer 11

yeah that is right

Answer 12

ok cool

Answer 13

looks like you got your work cut out for you

Answer 14

ya lol ty