Fermat's Last Theorem: Proof for n=7: Existence of v

In today's blog, I will continue the proof for Fermat's Last Theorem n=7.

The details for today are based on Paulo Ribenboim's Fermat's Last Theorem for Amateurs.

Lemma 1: Given u² = s⁴ + 6t²s² - t⁴/7 where s,t are integers and gcd(s,t)=1, we can conclude that there exists an integer v such that:
(a) t = 7^ev
(b) 7 doesn't divide v
(c) gcd(s² + 3*7^2ev² + u,s² + 3*7^2ev² - u) is a power of 2 or 1.

Proof:

(1) s,t are integers so 7s⁴ + 42t²s² - t⁴ is an integer.

(2) But this means that 7u² is an integer which means that u² is an integer. If u² were not an integer then it u = would have to equal some number of √7 which is impossible since u is a rational number and √7 is not (see here for details).

(3) But from (a), we also note that 7 must divide t⁴ which means from Euclid's Generalized Lemma, 7 must divide t.

(4) If e is the number of times that 7 divides t, then there exists v such that:
t = 7^ev and 7 doesn't divide v.

(5) Inserting this into the value of u² gives us:
u² = s⁴ + 6*(7^2e)v²s² - 7^4e-1v⁴ = (s² + 3*7^2ev²)² - 64*7^4e-1v⁴

(6) Subtracting u² from both sides and adding 64*7^4e-1v⁴ to both sides gives us:
(s² + 3*7^2ev²)² - u² = (s² + 3*7^2ev² + u)(s² + 3*7^2ev² - u) = 64*7^4e-1v⁴

(7) Now, we can show that gcd( s² + 3*7^2ev² + u , s² + 3*7^2ev² - u) = 2^f since:

(a) Assume that a prime p divides both and it is not 2.

(b) Then p divides u [since 2u = s² + 3*7^2ev² + u -( s² + 3*7^2ev² - u) and p doesn't divide 2.]

(c) We also know that p divides s² + 3*7^2ev² [since 2(s² + 3*7^2ev²) = s² + 3*7^2ev² + u + (s² + 3*7^2ev² - u) and p doesn't divide 2]

(d) We know that p ≠ 7 since then 7 would divide s but this is impossible since 7 divides t and gcd(s,t)=1.

(e) We know from #6, that p divides 64*7^4e-1v⁴

(f) This means that p must divide v (since we are assuming it is not 2 and we know it is not 7)

(g) Going back to (c), it must also divide s.

(h) But this is impossible since it would then also divide t (since t = 7^ev) and we know that gcd(s,t)=1.

(i) So we have a contradiction and we reject our assumption (7a).

QED