## Definition

Number theory is a branch of mathematics which helps to study the set of positive whole numbers, say 1, 2, 3, 4, 5, 6,. . . , which are also called the set of natural numbers and sometimes called “higher arithmetic”.

Number theory helps to study the relationships between different sorts of numbers. Natural numbers are separated into a variety of times.1

Number theory is the study of the integers (e.g. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis.2

Number theory is a vast and fascinating field of mathematics, sometimes called “higher arithmetic,” consisting of the study of the properties of whole numbers. Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor functionRiemann zeta function, and totient function. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (2005abc).

The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that “it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics.” Gauss, often known as the “prince of mathematics,” called mathematics the “queen of the sciences” and considered number theory the “queen of mathematics” (Beiler 1966, Goldman 1997). 3

## Who

What careers use number theory?

• Electrical Engineering
• Electronics
• Machine Learning
• Artificial Intelligence
• Programming Languages
• Molecular Biology
• Mechanical Engineering
• Systems Engineering

## What

The most important application of number theory is that it is the key foundation of cryptography. Our strong encryption algorithms and systems have developed because of the impetus provided by number theory. For example, your data cannot be easily accessed by anyone because of the strong encryption system. Moreover number theory is useful in the study of binary codes and other related concepts. This is the main real life applications of number theory. Other than those, there are myriad of applications which are useful for applied mathematicians and physicists. For example, the q series is extremely useful in the study of strings. Then supersymmetric functions like the mock theta functions and theta functions are used for a large number of advanced purposes. Over-all number theory is extremely useful for applied mathematicians and physicists. But to understand those applications you must have a sound knowledge in both. 4

Number theory is used to find out if a given integer ‘m’ is divisible with the integer ‘n’ and this is used in many divisibility tests. This theory is not only used in Mathematics, but also applied in cryptography, device authentication, websites for e-commerce, coding, security systems, and many more. 5

Very basic number theory (like modular arithmetic or finding the solution to a linear Diophantine equation), has a ton of applications, but mostly we do not even realize we are doing math, like when you need to decide how many 6-pack of frankfurters and how many 8-pack of hot dog buns you want to buy to have no orphans.

Intermediate number theory had found application in cryptography. 6

## How

Many of the References and Additional Reading websites and Videos will assist you with understanding number theory.

As some professors say: “It is intuitively obvious to even the most casual observer.

1 “Number Theory (Definition, Basics, Examples)”. 2022. BYJUS. https://byjus.com/maths/number-theory/.

2 “Number Theory | Department Of Mathematics”. 2022. math.duke.edu. https://math.duke.edu/research/number-theory.

3 “Number Theory — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/NumberTheory.html.

4 “Real-Life, Every Day Applications Of Number Theory?” 2021. Mathematics Stack Exchange. https://math.stackexchange.com/a/3975948.

5 “Number Theory – Definition, Examples, Applications”. 2023. CUEMATH. https://www.cuemath.com/numbers/number-theory/.

6 “Number Theory and its applications to the real world”. 2023. reddit.com. https://www.reddit.com/r/math/comments/fqeuto/comment/flq6mrs/.

“1.1: What Is Number Theory?”. 2022. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.01%3A_What_is_Number_Theory.

“1.4: Combinatorics and Number Theory”. 2022. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Combinatorics_(Keller_and_Trotter)/01%3A_An_Introduction_to_Combinatorics/1.04%3A_Combinatorics_and_Number_Theory.

“1.4: Definitions of Elementary Number Theory”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/01%3A_Introduction_and_Notation/1.04%3A_Definitions_of_Elementary_Number_Theory.

“2.1: A Taste Of Number Theory”. 2022. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/02%3A_New_Page/2.01%3A_New_Page.

“5: Number Theory”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Courses/College_of_the_Canyons/Math_130%3A_Math_for_Elementary_School_Teachers_(Lagusker)/05%3A_Number_Theory.

“5.1: Number Theory- Divisibility And Congruence”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts_-_The_Fundamentals_of_Abstract_Mathematics_(Morris_and_Morris)/05%3A_Sample_Topics/5.01%3A_Number_theory-_divisibility_and_congruence.

“5.2: Introduction to Number Theory”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/5%3A_Additional_Topics/5.2%3A_Introduction_to_Number_Theory.

“6: Number Theory”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Courses/Las_Positas_College/Math_27%3A_Number_Systems_for_Educators/06%3A_Number_Theory.

“6.2: Introduction to Number Theory”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Courses/Saint_Mary’s_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/6%3A_Additional_Topics/6.2%3A_Introduction_to_Number_Theory.

“8: Number Theory”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Understanding_Elementary_Mathematics_(Harland)/08%3A_Number_Theory.

“8: Other Topics In Number Theory – Mathematics LibreTexts”. 2023. math.libretexts.org. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/08%3A_Other_Topics_in_Number_Theory.

“8: Topics In Number Theory”. 2017. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/08%3A_Topics_in_Number_Theory.

“8.5: Applications in Number Theory”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts_-_The_Fundamentals_of_Abstract_Mathematics_(Morris_and_Morris)/08%3A_Proof_by_Induction/8.05%3A_Applications_in_number_theory.

“9: Number Theory”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Courses/Hartnell_College/Mathematics_for_Elementary_Teachers/09%3A_Number_Theory.

“An Introduction to Number Theory (Veerman) – Mathematics LibreTexts”. 2023. math.libretexts.org. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman).

“An Introduction to the Theory of Numbers (Moser)”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_the_Theory_of_Numbers_(Moser).

“C: Elementary Number Theory”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_(Clark)/02%3A_Appendices/2.03%3A_Elementary_Number_Theory.

“Elementary Number Theory (Barrus And Clark)”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark).

“Elementary Number Theory (Clark)”. 2021. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Clark).

“Elementary Number Theory (Raji)”. 2018. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji).

“Number Theory – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Number_theory.

Rothschild, Ephraim. “What Are The Applications Of Number Theory In Computer Science Other Than Programming Problems On Online Judges?” 2023. Quora. https://qr.ae/promxg.

“Yet Another Introductory Number Theory Textbook – Cryptology Emphasis (Poritz)”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz).

## Videos

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).