An Introduction to the Theory of Numbers #1 - Number Theory

Hello everyone!

So I'm back with more material and information on math for you all! I'm really excited about this!

So, we are starting with some posts on Number Theory - Introduction, Major Concepts, Famous Problems, Breakthroughs and Discoveries and Famous Scientists behind them. So let us begin this journey with an Introduction to Number Theory!

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"Die Mathematik ist die Königin der Wissenschaften und die Arithmetik die Königin der Mathematik"

- Carl Friedrich Gauss (The 'Prince' of Mathematics)

This famous quote by Carl Gauss, popularly known as the Prince of Mathematics, literally translates to - "Mathematics is the Queen of all sciences, and Number Theory is the Queen of Mathematics".

And this statement holds true even nearly 200 years since Gauss lived.

Number Theory is the most fascinating, intriguing and an evergreen field of mathematics. There are surprises at every corner and interesting discoveries every moment! We will have a look at what the essence of number theory lies in, in this blog.

Number Theory - Number Theory is the field of mathematics that involves the study of integers (mostly whole numbers) and their properties, as well as properties of all structures which require whole numbers as their basis.

As numbers are central to number theory, let us define some important sets:

The set of integers- Ƶ = {..., -2, -1, 0, 1, 2, 3, ...} 'Z' represents Zahlen, meaning Number

The set of whole numbers- W = {0, 1, 2, 3, ...}

The set of natural numbers- N = {0, 1, 2, 3, ...}

Also, we will be using some basic results as a foundation, some of which are -

• Fundamental Theorem of Arithmetic (https://brilliant.org/wiki/fundamental-theorem-of-arithmetic/)

• Principles of Divisibility of Numbers (Use of symbols such as a|b meaning a is a factor of b)

• Basic operations on Natural Numbers, including the laws of exponents (https://brilliant.org/wiki/exponential-functions-properties/)

Now let's move straight into the topic. We will have a look at some of the most basic truths that play a vital role in number theory as well as other topics in math.

Prime Numbers -

Let a be a positive integer. If b is a divisor of a that is not equal to 1 or a (i.e. b is a nontrivial proper divisor of a), then a can be factored as a=bc, where b and c are nontrivial proper divisors of a. Then we can break down b and c similarly to get an expression for aa as a product of smaller divisors. The process stops only when each of the divisors in the product cannot be broken down further; in other words, when the divisors in the product do not have any nontrivial proper divisors.

Definition - Integers greater than 1 with no nontrivial proper divisors are called prime numbers.

The distribution of primes inside the integers is a difficult and rich topic of research. There are infinitely many primes, a fact which was known to Euclid. A much more sophisticated estimate is the prime number theorem, which says roughly that the probability of a random integer ≤ x being prime is about 1/(ln x)​. Other, more precise estimates often involve the Riemann Hypothesis, a deep and unsolved conjecture about the zeroes of a certain complex-valued function.

Still, other elementary questions about the prime numbers remain open. In particular, the twin prime conjecture that there are infinitely many prime numbers p such that p+2 is also prime, and the Goldbach conjecture that any even integer ≥ 4 can be written as a sum of two primes, are still unsolved despite centuries of efforts by countless mathematicians

Modular Arithmetic -

Results involving divisibility are often most easily stated using modular arithmetic. If two integers a and b leave the same remainder when divided by an integer n, we write a ≡ b (mod n), read "a is congruent to b mod n." For example, 64 ≡ 1 (mod 7) as 64 = 7 x 9 + 1 leaves remainder 1 when divided by 7.

Basic rules of modular arithmetic help explain various divisibility tests learned in elementary school. For instance,

> Every positive integer is congruent 0 (mod 3) to the sum of its digits.

Some of the basic rules of modularity are -

The operations of addition, subtraction, and multiplication work as expected, but there is also a form of division. In particular, the equation x ≡ 1/a (mod n) makes sense if ax ≡ 1 (mod n). Such an x exists if and only if gcd (a,n) = 1, by Bezout's identity (https://brilliant.org/wiki/bezouts-identity/)

That's it for today; We will meet again in late August with the next post of this series!

See you all soon!

Stay safe!

Resources -

• Brilliant (https://brilliant.org/wiki/number-theory/)

• MathWorld @Wolfram (https://mathworld.wolfram.com/NumberTheory.html)

• Elementary Number Theory (by David Burton) (check out Maths Resources page of the website)

#Mathematics #NumberTheory #MathsPrime