After this proof, it became traditional to divide Fermat's Last Theorem into two parts:
Case I: None of the three values x,y,z is divisible by n.
Case II: n divides 1 of the three values x,y,z.
Here is Sophie Germain's Theorem:
Theorem: If xn + yn = zn and n is a prime ≥ 3 and 2n+1 is a prime, then n must divide xyz.
(1) We can assume that x,y,z are relatively prime to each other. [See here]
(2) So, let's assume that n doesn't divide xyz.
(3) Since n is odd, we can set z' to -z and get:
xn + yn + (z')n = 0.
(4) We can rewrite this to be [see here for details]
-xn = (y+z)(y(n-1) - y(n-2)z + ... - yz(n-2) + z(n-1))
(5) From this, we can show that:
y+z and y(n-1) - y(n-2)z + ... - yz(n-2) + z(n-1) are relatively prime since:
(a) Assume that they are not relatively prime.
(b) Then, there exists a prime p such that p divides y+z and p divides y(n-1) - y(n-2)z + ... - yz(n-2) + z(n-1).
(c) from this, we know that z ≡ -y (mod p).
(d) Using the fact that they are congruent (see here), we get:
y(n-1) - y(n-2)z + ... - yz(n-2) + z(n-1) ≡
y(n-1) - y(n-2)(-y) + ... -y(-y)(n-2) + (-y)(n-1) ≡
y(n-1) + y(n-1) + ... + y(n-1) + y(n-1) ≡ (n)y(n-1)
(e) So from Euclid's Lemma, p either divides n or it divides y(n-1).
(f) But p doesn't divide n since:
(i) n is prime so p would equal n (a prime is only divisible by itself or 1)(g) And p doesn't divide y(n-1) because then p would divide y (Euclid's Generalized Lemma) and p would also divide z (since it divides y and it divides y+z), but this is impossible by step #1.
(ii) And then n would divide (-x)n
(iii) Which means that it would divide x (from Euclid's Generalized Lemma)
(iv) But this impossible by step #2.
(h) So we have a contradiction and we reject our assumption.
(6) From Step #5 (see here), we can conclude that there exists a such that:
y + z = an
(7) But we can make this same argument for z+x and x + y where:
-yn = (x + z)(x(n-1) - x(n-2)z + ... -xz(n-2) + z(n-1))
-zn = (x + y)(x(n-1) - x(n-2)y + ... -xy(n-2) + y(n-1))
(8) So we know that there exists b,c where:
z + x = bn
x + y = cn
(9) We also know that any value un ≡ ±1 or 0 (mod 2n + 1) since:
(a) Assume that 2n+1 doesn't divide un [ we only need to consider the case where it is not congruent to 0]
(b) Since 2n+1 is a prime, we can apply Fermat's Little Theorem to get:
u2n ≡ 1 (mod 2n+1)
(c) And this means that:
(un)2 ≡ 1 (mod 2n + 1)
(d) So that un ≡ ± 1 (mod 2n+1) [See here for review of modular arithmetic]
(10) From this, we can conclude that (2n+1) divides xyz' since:
(a) Assume 2n+1 doesn't divide xyz'.
(b) Then, from step #9, we have:
xn ≡ ± 1
yn ≡ ± 1
zn ≡ ± 1
(c) But this is impossible since:
xn + yn + (-z)n ≡ 0 (mod 2n+1)
± 1 ± 1 ± 1 cannot be congruent to 0 (mod 2n+1)
(d) So, we reject(a).
(11) Let's assume that (2n+1) divides x. (We will be able to make the same type of argument if it divides y or z')
(12) Then 2x = bn + cn + (-a)n since from step #6 and step #8, we know that:
2x = z + x + x + y -y - z = bn + cn + (-a)n
(13) And we can conclude that (2n+1) divides acb since;
(a) We assumed that 2n+ 1 divides x. This gives us:
bn + cn + (-a)n ≡ 0 (mod 2n+1)
(b) Now, we can apply the same argument as Step #10 since:
(i) Assume that 2n+1 doesn't divide acb
(ii) From step #9 and (i) above, we can conclude that bn, cn, (-a)n ≡ ± 1.
(iii) But this is impossible since:
± 1 ± 1 ± 1 cannot be congruent to 0 (mod 2n+1)
(iv) So we can reject our assumption in (i) and conclude that 2n+1 divides acb.
(14) But (2n+1) can't divide abc since:
(a) (2n+1) can't divide b since:
(i) 2n+1 divides b → 2n+1 divides bn → 2n+1 divides z + x [since by step #8, z + x = bn.]
(ii) We know that 2n+1 divides x, so if it divides z+x, it must also divide z.
(iii) But this is impossible since x,z have no common factors by step #1.
(b) (2n+1) can't divide c for the same reason as above. That is, x,y have no common factors by step #1.
(c) (2n+1) can't divide a since:
(i) Assume (2n+1) divides a.
(ii) Then (2n+1) divides y + z which means that z ≡ -y (mod 2n+1)
(iii) We know from Step #5, there exists a value d, e such that:
dn = y(n-1) - y(n-2)z + ... - yz(n-2) + z(n-1).
en = x(n-1) - x(n-2)y + ... -xy(n-2) + y(n-1).
(iv) Applying (ii) gives us:
dn ≡ (n)y(n-1) (mod 2n+1) [See Step #5d]
(v) Since we know that x ≡ 0 (mod 2n + 1) [from Step #11], we get:
en ≡ (0)(n-1) - (0)(n--2)y + ... -(0)y(n-2) + y(n-1) ≡ y(n-1) (mod 2n+1)
(vi) Appyling (iv), this gives us:
dn ≡ (n)en (mod 2n+1)
(vii) From Step #9, we know that dn,en are 0,1, or -1. This means that they must both be equal to 0 since this is the only way that 0 ≡ n*0.
(viii) But if 2n+1 divides both dn,en, then it divides both d,e from Euclid's Lemma
(ix) But 2n+1 divides e implies that it also divides y (from (v).)
(x). But then it divides both x,y which goes against step #1.
(xi). So we have a contradiction and we reject (i).
(15) And this means that step #14 contradicts step #13 so we reject Step #2.
Corollary: x5 + y5 = z5 → 5 divides xyz.
This is true by above since 5 is a prime and 2*5+1 = 11 is a prime.