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)
Singapore Maths Tuition 
Proof of Fermat’s Last Theorem
a^n+b^n=c^n
odd+even=odd
a^3+b^3=c^3
let c=2x+1
8x^3+12x^2+6x+1=c^3
let b=2x
b^3+12x^2+6x+1=c^3
a^3=12x^2+6x+1
Only valid solution to a is when x=0 for all
c=2x+1,2x+3,2x+5,…
n=3,4,5,…
No whole number solution for a,b,c when n>2
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