From Little to Lost in the Margins: A Journey through Number Theory

By Aniya Thangamuthu

In the fascinating world of mathematics, there are those moments when a single equation grabs the attention of scholars for centuries. Fermat’s last theorem is one of them, with a unique sense of mystery as Fermat wrote similar to this line, ‘this proof is too large for the margin’, leaving mathematicians with a challenge, a riddle that captivated them for more than 300 years.

Fermat's Last Theorem and its precursor, Fermat's Little Theorem, are two such equations that have kicked off a remarkable journey through the intricacies of mathematics. As we embark on this intellectual adventure, let's unravel the profound ideas that both these theorems have sparked, expanding our grasp of numbers and equations along the way.

Fermat's Little Theorem: More than its name suggests

Before we dive headfirst into Fermat's Last Theorem, it's crucial to get acquainted with Fermat's Little Theorem. Pierre de Fermat, a 17th-century French maths whiz, left behind some cryptic notes in the margins of his copy of Arithmetica. One of these notes would later become known as Fermat's Little Theorem. 

So, what's the deal with Fermat's Little Theorem? It's quite simple. If you've got a prime number 'p' and any integer 'a' that's not divisible by 'p,' then here's what you've got:

In this equation, "≡" denotes congruence, indicating that a(p-1) leaves the remainder, 1 each time when divided by 'p.' Despite its seemingly straightforward nature, this theorem stands as a fundamental result in number theory and serves as a prelude to the more intricate concepts we will explore.

Moreover, Fermat's Little Theorem plays a significant role in various cryptographic protocols, ensuring the security of data transmission by providing a reliable means to verify the primality of numbers. It is instrumental in the encryption algorithms safeguarding our digital communications.

Fermat Last Theorem: The History behind it all

Fermat first mentioned the theorem in the margin of his copy of an ancient Greek text known as Arithmetica. He wrote that he had discovered a "truly marvellous proof" for this proposition but did not provide any details. Fermat's note remained a tantalising mystery for mathematicians for many years, sparking the curiosity among them.

This proof is a story of intellectual grit and genius. Pierre de Fermat's scribbles in the margins of Arithmetica set the stage for over three centuries of mathematical curiosity.

This theorem is like the riddle of the Sphinx. It says that for any three positive integers 'a,' 'b,' and 'c' greater than zero, the equation an + bn = cn has no positive integer solutions for 'n' greater than 2. This seemingly simple statement sparked a pursuit that involved some of the brightest minds in mathematics.

At just nine years old, Andrew Wiles, stumbled upon Fermat's Last Theorem in his local library, sparking a lifelong commitment to solve it. He worked independently for decades, facing both challenges and personal tenacity. After unveiling his proof to the world, a critical error was discovered, creating a setback. Undeterred, Wiles remained steadfast in his pursuit, diligently revising his work. His enduring determination and dedication serve as a testament to the power of human perseverance.

Modular Forms: Patterns of Numbers

Our journey continues with modular forms, a mathematical concept that intersects with both Fermat's Last Theorem and Fermat's Little Theorem in intriguing ways. 

Modular forms are mathematical functions that have a special kind of symmetry and transform predictably under certain mathematical operations. These functions are important in various areas of mathematics, particularly in number theory, algebraic geometry, and complex analysis. 

Modular arithmetic, often introduced using the analogy of a clock face, involves considering numbers in relation to a modulus, where numbers "wrap around" after reaching a certain value. The notation a≡b mod n signifies that a and b leave the same remainder when divided by n. Useful properties include adding and multiplying congruences. When working with primes, each non-zero number has a multiplicative inverse mod p, providing a powerful tool. The concept is applied in divisibility tests, such as the one for determining if a number is divisible by 3 by examining the sum of its digits.

In number theory, Modular forms play a significant role in studying properties of whole numbers, such as prime numbers. They are also essential in understanding elliptic curves, which are used in cryptography and have applications in modern data security.

An elliptic curve, defined by a cubic equation in two variables, is not actually an "ellipse" in the traditional sense, but more like an abstract geometric shape defined by a specific mathematical equation. The equation defining an elliptic curve has the form: y2 = x3 + ax + b, where a and b are constants. On an elliptic curve, you can define points that satisfy the curve's equation. These points have both x and y coordinates. You can also define operations on these points, like addition and doubling. When you define these operations, the set of points on an elliptic curve forms a mathematical group, which means it has certain algebraic properties. The addition and doubling operations are defined so that they preserve this group structure.

Elliptic curves, closely intertwined with modular forms, serve as a bridge connecting algebraic geometry and number theory. These curves are defined by cubic equations and are characterised by their elegant geometric properties. In the context of Fermat's Last Theorem, elliptic curves became a pivotal tool in Andrew Wiles' groundbreaking proof.

Taniyama-Shimura-Weil Conjecture: A Bridge Between Worlds

The connection between Fermat's Last Theorem and modular forms deepens with the Taniyama-Shimura-Weil conjecture. This conjecture, proposed by Japanese mathematician Goro Shimura and French mathematician André Weil, proposes a profound link between elliptic curves and modular forms.

It suggests that every elliptic curve has a modular form. It’s like a cosmic connection between two distant galaxies. This conjecture sat there, teasing mathematicians for years, until Andrew Wiles and Richard Taylor finally proved it.  This proof was a pivotal step toward the confirmation of Fermat's Last Theorem.

Euler's Equation: A Harmonic Convergence

As we dive deeper into the wonders of the mathematical world with Fermat's Last Theorem and Fermat's Little Theorem, we bump into Euler's equation. This equation is a masterpiece of the maths world, and it goes like this:

It’s a mind-bender that links five of the most fundamental constants in maths: e, i, π, 1, and 0. Bringing everything together, in such an elegant and simple way.

This equation shows the deep interconnections between complex numbers, exponential functions, trigonometry, and arithmetic. While it is not directly related to Fermat's Last Theorem or Fermat's Little Theorem, it emphasises the profound elegance and unity that mathematics can achieve. Euler's equation serves as a reminder of the intricate interconnectedness of mathematical concepts and their capacity to inspire awe and curiosity, which I hope you feel too.

Fermat's Last Theorem: A Monument of Mathematical Inquiry

The grand finale came in 1994 when Andrew Wiles delivered the proof. It was a symphony of mathematical concepts – from elliptic curves and modular forms to the Taniyama-Shimura-Weil conjecture. Wiles' achievement showcased the power of teamwork in mathematics and cemented Fermat's Last Theorem as a cornerstone of number theory.

To conclude, Fermat's Last Theorem and Fermat's Little Theorem have kept mathematicians on their toes for centuries, and they've unveiled profound mathematical concepts. From modular forms to the Taniyama-Shimura-Weil conjecture and Euler's equation, the world of maths keeps expanding, revealing how it's a labyrinth, full of answers and questions.

As we journey through this mathematical odyssey, we're reminded that maths isn't just a subject; it's a boundless adventure. These mathematical ideas highlight the beauty and unity of the mathematical universe, inviting us to explore its mysteries further. 

Extra reading:

If you want to understand Fermat’s Last Theorem better, I would recommend reading the book Fermat’s Last Theorem by Simon Singh, really fun to read!

Or would prefer to watch something, here is the author on the background of the theorem on the famous maths channel, Numberphile: https://youtu.be/qiNcEguuFSA

Here’s a link to a more in depth introduction to modular arithmetic if you never heard of it before: https://nrich.maths.org/4350