Shimura-Taniyama-Weil Conjecture (Modularity Theorem)

Shimura and Tanyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.

The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.

It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:

Example:
$latex boxed {y^{2} + y = x^{3} – x^{2}
} &fg=aa0000&s=3 $ — (I)

There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.

For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)

Note: the last solution when y=4,
Left side = 16 + 4 = 20 = 4×5 = 0…