D <=> A
D : Diaphantine equations
eg.
A = Automorphic Form (eg. Elliptic Curves used in highest Cryptography “ECC” )
Fermat’s Last Theorem: after failures of 350 yrs, Andrew Wiles in 1994 proved easier from reversed direction A to D.
Lycée Pierre de Fermat (Maths Supérieures & Spéciales @Toulouse, France) was my Alma mater。
https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/
Use Fermat’s Last Theorem to prove irrational of cubic root of 2.
… (Read on) from source :
Summary:
Andrew Wiles’ Proof of Fermat’s Last Theorem (FLT) by contradiction :
A. Assume FLT is true for all prime p (Why? sufficient to prove only for prime) such that:
B. then a, b, c could be rearranged into an Elliptic Curve,
C. then leverage such Elliptic Curve into a Galois Represebtation.
D. then a Modular Form.
E. then leads to an impossible weight 2 level 2 Modular Form.
Hence,
¬E -> ¬D -> ¬C -> ¬B -> ¬A (proved)
1950s Taniyama-Shimura-Weil proved the link below:
B -> (via assume C) -> D
Andrew Wiles’ took 7 years to complete the whole proof in 1994 by proving the missing link C -> D.
Assume
![\sqrt [3] 2](https://s0.wp.com/latex.php?latex=%5Csqrt+%5B3%5D+2&bg=ffffff&fg=375362&s=0 "\sqrt [3] 2")
![\displaystyle \sqrt [3] 2 = \frac {q}{p} \,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csqrt+%5B3%5D+2+%3D+%5Cfrac+%7Bq%7D%7Bp%7D+%5C%2C+&bg=ffffff&fg=375362&s=0 "\displaystyle \sqrt [3] 2 = \frac {q}{p} \,")
After arrangement, we get
According to Fermat’s Last Theorem, the equation [*] has no solution.
Hence, par absurd,
I am a fan of Fermat, not only because my university Alma Mater was in his hometown Toulouse (France) named after him “Lycée Pierre de Fermat (Classe Préparatoire Aux Grandes Ecoles) ” , but also the “Fermat’s Last Theorem” (FLT) has fascinated for 350 years all great Mathematicians including Euler, Gauss,… until 1993 finally proved by the Cambridge Professor Andrew Wiles. Another “Fermat’s Little Theorem” is applied in computer Cryptography .
Below is the explanation of (n = 4) case proved by Fermat and the latest proof by contradiction.
Euler Conjecture: extends FLT to 4 or more integers if FLT still holds? (a contradiction found).
Simpsons “Fool” Equality: Proof by contradiction (odd = even)
Proof of FLT by Andrew Wiles (1993):
The proof by Contradiction of FLT (n=4) is in Part 2 of the video after 20:30 mins (Warning: a bit heavy)
How likely is it that a mathematics student can’t solve IMO problems?
Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh
For all x, y, z integers,
Proof:
Reduce n to 2 categories:
1. n |4:
Fermat proved n=4.
2. n not |4 => n|p, (p odd prime).
n=3 proved since 1770.
Conclude:
Prove FLT no solution for (n> 2) <=> Prove for (odd primes p≥ 5)
