number theory
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What is number theory?
Number theory, sometimes called "higher arithmetic," is a branch of mathematics that explores the properties and relationships of positive integers. Once considered a pure area of mathematics with no real-world applications, it now plays a vital role in modern digital communications and computing.
Modern number theory is broad, with sub-classifications such as elementary, algebraic, analytic, geometric, and probabilistic number theory. These reflect the varied approaches to addressing problems related to integers.
The work of Carl Friedrich Gauss in Disquisitiones Arithmeticae (1801) was highly significant to the field, becoming a foundation for number theory. In it, Gauss organized the work of his predecessors and provided the first modern proof of the unique factorization theorem.
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What is a prime number?
A prime number is a positive integer greater than 1 that can only be divided evenly by 1 and itself. For example, 2, 3, 5, 7, and 11 are prime numbers, while 4, 6, and 9 are not because they are divisible by other numbers.
Primes are the multiplicative building blocks for all natural numbers greater than zero. Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers.
Prime numbers have been studied since antiquity. Euclid proved that there are infinitely many primes. Today, mathematicians continue to study prime numbers and their properties.
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How can you tell if a number is even or odd?
To determine if a whole number is even or odd, see if the number can be divided by 2. If it can be divided by 2 without a remainder, then it is an even number. If there is a remainder after dividing by 2, then it is an odd number. For example, 10 ÷ 2 = 5, so 10 is an even number. 101 ÷ 2 = 50.5, so 101 is an odd number. Zero is divisible by 2, so zero is an even number.
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What are factors and multiples in number theory?
In number theory, factors are numbers or algebraic expressions that divide another number or expression evenly, leaving no remainder. For instance, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 and 12 ÷ 6 = 2, without any remainder.
A multiple is the product of a quantity by an integer. For example, 35 is a multiple of 7.
Every composite number can be expressed as a product of prime numbers, and this representation is unique, except for the order in which the factors are written. The greatest common divisor (GCD) is the largest integer that divides each number in a set, while the least common multiple (LCM) is the smallest positive integer that is a multiple of each of the numbers.
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How do you find the greatest common factor (GCF) of two numbers?
The greatest common factor (GCF) is the largest integer that divides each number in a set of positive integers.
You can find the GCF of two numbers using these methods:
* Prime Factorization: Break down each number into its prime factors. Then, identify the common prime factors between the two numbers. Multiply those common prime factors, using the lowest power of each prime that appears in both factorizations.
* Euclidean Algorithm: The Euclidean algorithm is a method that can find the greatest common divisor (GCD) of two numbers. The algorithm involves successively dividing and calculating remainders. Continue the division process until the remainder is zero; the last positive remainder is the GCD of the original two numbers.3 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
What does it mean for two numbers to be relatively prime?
In number theory, two numbers are relatively prime if they share no common factors other than 1.
For a set of positive integers ( a_1, a_2, ..., a_k ), their greatest common divisor (GCD) is the largest integer that divides all of them. If their GCD is 1, the numbers are said to be relatively prime. For example, 12 and 25 are relatively prime because they have no common factors other than 1.
The Euclidean algorithm is a method to find the GCD of two numbers without factoring them into primes.
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number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural of mathematical pursuits.
Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background.
Until the mid-20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. The advent of digital computers and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes, testing conjectures, and solving numerical problems once considered out of reach.
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Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers.
From prehistory through Classical Greece
The ability to count dates back to prehistoric times. This is evident from archaeological artifacts, such as a 10,000-year-old bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something. Very near the dawn of civilization, people had grasped the idea of “multiplicity” and thereby had taken the first steps toward a study of numbers.
It is certain that an understanding of numbers existed in ancient Mesopotamia, Egypt, China, and India, for tablets, papyri, and temple carvings from these early cultures have survived. A Babylonian tablet known as Plimpton 322 (c. 1700 bce) is a case in point. In modern notation, it displays number triples x, y, and z with the property that x2 + y2 = z2. One such triple is 2,291, 2,700, and 3,541, where 2,2912 + 2,7002 = 3,5412. This certainly reveals a degree of number theoretic sophistication in ancient Babylon.
Despite such isolated results, a general theory of numbers was nonexistent. For this—as with so much of theoretical mathematics—one must look to the Classical Greeks, whose groundbreaking achievements displayed an odd fusion of the mystical tendencies of the Pythagoreans and the severe logic of Euclid’s Elements (c. 300 bce).
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Pythagoras
According to tradition, Pythagoras (c. 580–500 bce) worked in southern Italy amid devoted followers. His philosophy enshrined number as the unifying concept necessary for understanding everything from planetary motion to musical harmony. Given this viewpoint, it is not surprising that the Pythagoreans attributed quasi-rational properties to certain numbers.
For instance, they attached significance to perfect numbers—i.e., those that equal the sum of their proper divisors. Examples are 6 (whose proper divisors 1, 2, and 3 sum to 6) and 28 (1 + 2 + 4 + 7 + 14). The Greek philosopher Nicomachus of Gerasa (flourished c. 100 ce), writing centuries after Pythagoras but clearly in his philosophical debt, stated that perfect numbers represented “virtues, wealth, moderation, propriety, and beauty.” (Some modern writers label such nonsense numerical theology.)
In a similar vein, the Greeks called a pair of integers amicable (“friendly”) if each was the sum of the proper divisors of the other. They knew only a single amicable pair: 220 and 284. One can easily check that the sum of the proper divisors of 284 is 1 + 2 + 4 + 71 + 142 = 220 and the sum of the proper divisors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. For those prone to number mysticism, such a phenomenon must have seemed like magic.