Number theory
𝑎𝑧+𝑏 𝑐𝑧+𝑑 , where a, b, c, d are whole numbers that satisfy the condition 𝑎𝑑 − 𝑏𝑐 = 1. In other
number of the form
words, the function is symmetric under these modular transformations.
ii) Holomorphic and well-behaved The function must be holomorphic, meaning it is smooth and differentiable at every point in the upper half of the complex plane. It must also behave nicely at the boundary point at infinity — specifically, it can grow, but only in a controlled way, not exploding wildly.
To put the prerequisites into more layman terms, it is a specific type of function exhibiting a high level of symmetry, such that transformations to the function become predictable. Now we have required knowledge to outline Wiles’ proof.
It was a proof by reductio ad absurdum, 1 or proof by contradiction. A few years prior to Wiles’ discovery, German mathematician Gerhard Frey proposed a new method of expressing elliptic curves, in the form:
𝑦 2 = 𝑥(𝑥−𝑎 𝑛 )(𝑥+𝑏 𝑛 )
What was different was that by having the terms 𝑎 𝑛 and 𝑏 𝑛 in the equation, it could be attached to any ‘Fermat equation’. Experts later confirmed that for any 𝑛 >2 , due to the curve’s arithmetic properties, it was unable to be expressed in a modular form. From 1986-1993, Wiles worked in secrecy to solve the Taniyama-Shimura Conjecture alone and complete the link between the 2 unsolved problems. After 7 years of restless nights, the proof was completed.
As shown in the Taniyama-Shimura Conjecture, rational elliptic curves must be expressed in a modular form. This directly contradicted what was proved above, and therefore the equation
𝑎 𝑛 +𝑏 𝑛 = 𝑐 𝑛
did not hold for all 𝑛 >2 . The problem that baffled mathematicians for four centuries was ultimately solved.
The Collatz Conjecture
We end the study with a much simpler problem. Proposed by German mathematician Lothar Collatz in 1937, its set up is extremely trivial to understand:
Choose any positive integer 𝑎 0 ; If 𝑎 𝑛 is odd, 𝑎 𝑛+1 =3𝑎 𝑛 +1; If 𝑎 𝑛 is even, 𝑎 𝑛+1 =
1 2
𝑎 𝑛
1 https://iep.utm.edu/reductio. Latin for ‘proof by contradiction’.
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