What Is Number Theory?

Anyone who's experienced love will tell you that it's the little quirks that truly matter. It's the inside jokes exchanged at the end of a long day. The unique way someone enjoys their morning coffee. The pile of dog-eared novels that accumulate beside their bed. These small, interconnected details become part of who we are. They outline the subtle layers of our character, and, to those who truly care, they reveal the deeper beauty beneath.

For some, the highest form of beauty resides within mathematics. They view the realm of numbers and, much like you wouldn't define your loved one by just their job or hair color, the math enthusiast sees beyond the surface of numbers. Figures like 6, 28, and 496 transform into more than mere carriers of data. They are transcendent, becoming more than their function, with their relationships showing the intricate structure of the universe itself.

The exploration of these hidden and profound relationships is known as number theory, sometimes called higher arithmetic. Number theorists examine the properties of integers, the basic numbers you know like -1, -2, 0, 1, 2, and so on. It's both a theoretical and experimental field, where mathematicians uncover surprising and fascinating mathematical connections.

Different types of relationships can be found among integers. We classify them based on their characteristics. For example, odd numbers (like 1, 3, 5 …) can't be evenly divided, while even numbers (such as 2, 4, 6 …) can. Then there are square numbers, which result from multiplying a number by itself. Take 2 x 2 = 4 and 3 x 3 = 9—both 4 and 9 are square numbers. This applies to 1 (1 x 1 = 1) and 9,801 (99 x 99 = 9,801) as well. These examples can be expressed as 2, 3, 1, and 99.

Now, let's deepen our exploration. In certain situations, adding square numbers together can produce new square numbers, which is known as a Pythagorean triple, adhering to the Pythagorean theorem (a + b = c). A prime example of this is 3 + 4 = 5, forming the set 3, 4, 5.

Number theory is all about exploring the relationships between numbers and posing new questions about them. But what exactly is a theory of numbers? How do mathematicians construct proofs, and why do some problems in math remain unsolved for centuries?

Questions in Number Theory

In the realm of mathematics, there are countless types of numbers, each possessing unique properties. Mathematicians craft theories about the connections between numbers and groups of numbers. These theories are supported by axioms (previously proven truths) and theorems (statements derived from other axioms or theorems).

The initial step in crafting a vibrant new mathematical theory is posing a theoretical question about how numbers relate to one another. For instance, could the sum of two cubes equal another cube? Recall the Pythagorean triples mentioned earlier—sets of three numbers like (3, 4, 5) that satisfy the equation a + b = c. But what if we consider a + b = c? In 1637, mathematician Pierre de Fermat raised a similar question about cubes and claimed to have devised a mathematical proof demonstrating, through a series of logical steps, that no, the sum of two cubes cannot form a cube. This became known as Fermat's Last Theorem. Sadly, Fermat never included the full proof in his notes, leaving behind only the famous statement, "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain" [source: NOVA].

For over three and a half centuries, mathematicians worldwide attempted, unsuccessfully, to uncover Fermat's proof. What was at stake in this pursuit? Aside from academic pride, it was driven by a passion for pure, abstract mathematics. Then, in 1993, thanks to advancements in computational mathematics that did not exist in Fermat's era, British mathematician Andrew Wiles finally proved the 356-year-old theorem. The debate still lingers over whether Fermat could have truly constructed such a profound proof in his pre-computer age, or if he was simply mistaken.

Number theory involves exploring various perceived or theoretical patterns in numbers or groups of numbers. This investigation begins with a vital component of intelligent thinking: pattern recognition. Joseph H. Silverman, a mathematics professor at Brown University, outlines five essential steps in number theory:

Thus, Fermat's Last Theorem remained a conjecture for 356 years, only becoming a proven theorem in 1993. On the other hand, Euclid's Proof of Infinite Primes (which demonstrates that prime numbers are infinite) has stood as a prime example of mathematical logic since 300 B.C. Some number theory conjectures, old and new, continue to lack a formal proof.

Numbers are as boundless as human comprehension is limited, ensuring that number theory and its various branches will continue to intrigue math enthusiasts for centuries. While some age-old problems may be solved, new and even more complex conjectures will emerge in their place.