Number Theory : The Queen of Mathematics

The Prince of Mathematics, Carl Friedrich Gauss once famously said :

Mathematics is the Queen of the Sciences and Number theory is the Queen of Mathematics.

This statement though highly debatable, does find resonance in many of the great mathematicians of both the past and the present (and most probably this will be true of the future too).  What is the reason for such ubiquitous fondness and high regard towards the theory of numbers? This question could be well put to rest by Erdös’ response:

It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”

However, I will attempt to give a sense of why this maybe so. The best way I felt to do this was to primarily write from my own experience about the reasons for my love of number theory. Along the way, I will give the most basic results and questions of elementary number theory which can go to further illustrate the inherent beauty of the field.

Everyone makes their first foray into Mathematics through interaction with the natural numbers, the fundamental building blocks which when extended lead to the integers and finally generalize to the complex numbers. The elementary theory of numbers studies one the most fundamental properties of the integers: divisibility. Though that may sound to be too basic, that is where the deceptive beauty of number theory comes in. Often the problems posed in number theory can be understood even by the layman but the solutions to these questions often require sophisticated reasoning and mathematical tools. This is what really appealed to me as an amateur; so basic yet so challenging. Below is a standard question from number theory, which illustrates the above fact.

Problem: Prove that if n is an integer greater than 1, then n cannot be a divisor of $2^n -1$.

The interested reader should definitely try the above question and a solution of this is available here.

The above solution also illustrates an important point: you have to sometimes approach a problem in an indirect manner, proving a related theorem which when applied repeatedly rapidly yields the solution. This modular way of thinking in number theory is often required and yields many beautiful solutions.

The use of number theory principles don’t have to necessary be circumscribed explicitly to integers. It pops up in the most unexpected places.  This feature of number theory is also one of the reasons for its foundational position in Mathematics. Take for example the following question about a rational number. Even though it is about fractions, number theory can be effectively used to arrive at the solution.

Problem : Find all positive rational numbers with the property that the sum of the number and its reciprocal is an integer.

A solution is available here. An algebraic approach is also possible but that is not as rapid as the number theoretic approach. At any rate, all solutions are important since each reveals something novel about the question and illuminates it in new light. A perfect case in point here is the important Law of Quadratic Reciprocity, the “golden theorem” of number theory, for which Gauss had at least six different proofs!

The fact that divisibility is one of the most significant  and beautiful concepts in number theory is reflected by the fact that primes continue to occupy centre-stage of number theory research. Primes are, in the language of chemistry, the building atoms of all integers. Primes are those natural numbers greater than 1 which have no positive divisors except 1 and p. Primes are, in the language of chemistry, the building atoms of all integers. To see why primes are such fundamental characters, consider the following theorem:

The Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely expressed as a product of primes, up to the order of the primes.

Incidentally, the reason why 1 is not considered a prime is to keep the above important theorem valid. For if 1 was to be considered a prime then the prime factorisation of any number would not be unique( $12 = 2^2.3= 2^2.3.1 = 2^2.3.1^2$).

What remains so mysterious about the primes is that a formula to exactly calculate the $n^{th}$ prime still evades capture from the sharpest attempts of mathematicians. This somehow seems to embody for me a deep mystery which will reveal itself when the time is ripe. Anyhow, it is clear as of now that significant advances in current mathematical knowledge will be needed to decode this baffling mystery.

It is worth noting here that many advances have been made which do begin to answer the prime number mystery. One of the most famous of these is the Prime number theorem. It provides an approximation of the distribution of primes which was independently proved by Jacques Hadamard and Charles Jean in the 19th century.

Below $\pi(x)$ denotes the number of primes not exceeding x. For example $\pi(4) = 2, \pi(10) = 4 \text{ and } \pi(100) = 25$.

Prime Number Theorem:  $\lim \limits_{x \rightarrow \infty}{}\frac{\pi(x)}{x/\log(x)} = 1$.

This statement implies that for large x, the number of primes lesser than or equal to x will be approximately equal to $\frac{x}{\log(x)}$.

Now intuitively, it seems that the number of primes should be infinite and this indeed is true. There are many beautiful proofs of the infinitude of primes; Euclid’s proof of this in his Elements has become a classic due its simple yet beautiful logic. However, here I want to reproduce another proof that I found was very elegant:

There are many other proofs of  the infinitude of primes; surprisingly there is an analytical proof and one other which uses topology ! The interested reader can refer to the book “Proofs from THE BOOK”(there is an interesting reason behind this intriguing title), which has the six famous proofs of the infinitude of primes.

The fact that a proof in number theory exists that uses topology, says something about the foundational role of number theory. This is also reflected by many elementary concepts that exist in number theory.

Let me exemplify this with the concept of congruences and its beautiful generalization in group theory.

$a \equiv x$ (mod n) implies that n divides $a-x$. This relation it turns out is a perfect example of what is called an equivalence relation which is an extremely important concept. An equivalence relation $\otimes$ on a set $\mathbb{S}$ satisfies the following conditions for all $a,b,c \in \mathbb{S}$:

1. $a \otimes a$
2. $a\otimes b \:implies \: b\otimes a$
3. $a\otimes b \: and \: b\otimes c \: imply \: a\otimes c$.

Now verifying that $a \equiv x$ (mod n) does indeed define an equivalence relation on the set of integers :

The first condition is clearly satisfied since $a \equiv a$(mod n),  $\forall a \in \mathbb{Z}$.

Also if $a \equiv b\:(mod \: n)$ then $b \equiv a\:(mod \: n)$ .

Finally, if $a \equiv b\: (mod \: n)$ and $b \equiv c\:(mod \: n)$, then $a \equiv c\:(mod \: n)$.

Thus all three conditions for an equivalence relation are satisfied by a congruence relation. This is significant because all the properties of an equivalence relation now apply to a congruence relation! This connection can be further developed by considering groups and their cosets, however, that would be too great a digression. It is worth noting that many theorems belonging to number theory like that of Fermat(not the last one but the the little one!) can be very easily established using group theory.

Many more connections with number theory may be established which will go to further reveal the beauty of number theory, however I leave that for a post for some other day!