Elementary number theory

originally forked from the elementary number theory section in the number theory article

Elementary number theory deals with the topics in number theory by means of basic methods in arithmetic. Its primary subjects of study are divisibility, factorization, and primality, as well as congruences in modular arithmetic. Other topic in elementary number theory are Diophantine equations, continued fraction, integer partitions, and Diophantine approximations.

Definition

More specifically, elementary number theory works with elementary proofs, a term that excludes the use of complex numbers but may include basic analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous. For example, proofs based on complex Tauberian theorems, such as Wiener–Ikehara, are often seen as quite enlightening but not elementary despite using Fourier analysis, not complex analysis. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a more advanced proof.

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithms. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed the product, such as $2\times 3=6$.

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, many of the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.

Key concepts

Divisibility, primality, factorisation

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithms. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed the product, such as $2\times 3=6$. Division is the inverse function of multiplication, for example $6÷2=3$.

Divisibility is a property between two nonzero integers related to division. Euclid's division lemma states that any integer $a$ and positive integer divisor $b$ can be written as $a=bq+r$, where the remainder $r<b$ accounts for the left over quantity. The number $a$ is said to be divisible by $b$ if $a$ is divided evenly by $b$ without remainder; that is, if $r=0$. An equivalent formulation is that $b$ divides $a$ and is denoted by a vertical bar, which in this case is $b|a$. Elementary number theory studies divisibility rules in order to quickly identify if a given integer is divisible by a fixed divisor. For instance, it is known that any integer is divisible by 3 if its digit sum is divisible by 3.

Elementary number theory studies different classes of integers such as even and odd numbers and prime numbers. They are well-defined in the ring of integers. For example, odd numbers are integers that leave a remainder when divided $2$. Formally, an odd number has the form $2k+1$ for some integer $k$. Conversely, even numbers are always divisible by $2$ A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. There are infinitely many prime numbers. A positive integer greater than 1 that is not prime is called a composite number.

Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition of an integer into a product of integers. The unique factorization theorem is the fundamental theorem of arithmetic that relates to prime factorization, integer factorization into a product of prime numbers. The theorem states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example, $120$ is expressed uniquely as $2\times 2\times 2\times 3\times 5$ or simply $2^3\times 3\times 5$.

Two integers have divisors and multiples in common. relations between integers in regard to division or multiplication. These concepts can be extended to more than two integers, where they must fulfil their relation with all integers. The greatest common divisor (gcd) is the largest of such divisors. Two integers are said to be coprime or relatively prime to one another if their greatest common divisor, and simultaneously their only divisor, is 1. The Euclidean algorithm computes the greatest common divisor of two integers $a,b$ by means of repeatedly applying the division lemma and shifting the divisor and remainder after every step.

Congruence

Modular arithmetic works with finite sets of integers and introduces the concepts of congruence and residue classes. A congruence of two integers $a,b$ modulo $n$ (a positive integer called the modulus) is an equivalence relation whereby $n|(a-b)$ is true. Performing Euclidean division on both $a$ and $n$, and on $b$ and $n$, yields the same remainder. This written as $a\equiv b(\mathrm{mod}n)$. In a manner analogous to the 12-hour clock, the sum of 4 and 9 is equal to 13, yet congruent to 1. A residue class modulo $n$ is a set that contains all integers congruent to a specified $r$ modulo $n$. For example, $6\mathbb{\mathbb{Z}}+1$ contains all multiples of 6 incremented by 1. Modular arithmetic provides a range of formulas for rapidly solving congruences of very large powers. An influential theorem is Fermat's little theorem, which states that if a prime $p$ is coprime to some integer $a$, then $a^{p-1}\equiv 1(\mathrm{mod}p)$ is true. Euler's theorem extends this to assert that every integer $n$ satisfies the congruence$$a^{\phi (n)}\equiv 1(\mathrm{mod}n),$$where Euler's totient function $\phi$ counts all positive integers up to $n$ that are coprime to $n$. Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra, such as the Chinese remainder theorem.

Diophantine equation

The algorithm can be extended to solve a special case of linear Diophantine equations $ax+by=1$. A Diophantine equation has several unknowns and integer coefficients. Another kind of Diophantine equation is described in the Pythagorean theorem, $x^2+y^2=z^2$, whose solutions are called Pythagorean triples if they are all integers.

Others

Another kind of expression is the continued fraction, which writes a sum of an integer and a fraction whose denominator is another such sum.

History

The knowledge of numbers existed in the early civilisations of Mesopotamia, Egypt, China, and India. Surviving sources take the form of tablets, papyri, and carvings. The development of number theory occurred independently and a systematic study did not exist. The first positional numeral system was developed by the Babylonians starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient. They may have been familiar with prime factorisation.

A famous Babylonian artefact of number theory is Plimpton 322. It is a fragment of a larger clay tablet, dated around 1800 BC, that contains a list of fifteen Pythagorean triples. A Pythagorean triple are three integers $a,b,c$ that satisfy the Pythagorean equation $a^2+b^2=c^2$. Their study likely initiated with the observation of the triple (3, 4, 5). The Babylonians recognised the connection between the equation and right triangles before Pythagoras did. There is no academic consensus on the method of generation, purpose, and author. The triples are too large for the method to have been generated by brute force. Its purpose is theorised to be related to number theory, trigonometry, or astronomy. Following the trigonometric interpretation, some historians have perceived it as a teacher's catalogue of parameters.

Ancient Egyptian arithmetic took an additive approach to multiplication and division. They had a method of representing fractions as sums of distinct unit fractions. The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. Around 300 BC, they knew a formula for the sum of triangular numbers. Their awareness of Pythagorean triples is less certain. A problem in the Berlin Papyrus 6610 #1 involves finding integer terms that fulfil a Pythagorean equation, although the solution is a multiple of (3, 4, 5).

While early civilizations primarily used numbers for concrete practical purposes like commercial activities and tax records, ancient Greek mathematicians began to explore the abstract nature of numbers. Numerology motivated mathematicians from different cultures to study integer properties. Their work survives in the form of book fragments and iterative copies.

They formalised number theory as a field of study, establishing core concepts such as divisibility, factorisation, the greatest common divisor, and Diophantine equations. A further contribution was their distinction of various classes of numbers, such as even numbers, odd numbers, and prime numbers. This included the discovery that numbers for certain geometric lengths are irrational and therefore cannot be expressed as a fraction of integers.

The works of philosophers Thales of Miletus and Pythagoras in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics. Pythagoras founded the school of thought that sought to understand number theory, geometry, astronomy, and music. Believing that everything revolves around numbers, the Pythagoreans tended to assign them mystical properties. They visualised numbers as amounts of pebbles and studied figurate numbers based on their arrangement. This included triangular, square, and pentagonal numbers. They also distinguished between classes such as odd and even, perfect, and amicable numbers.

Euclid's works shifted towards a rigourous number theory.

Euclid (c. 300 BC) compiled in his Elements contemporary knowledge of geometry and number theory. He incorporated earlier Pythagorean studies of integer properties. In contrast to the Pythagorean mysticism, he used formal proofs to establish mathematical truths and validate theories, a novel feature in ancient Greek mathematics. He ordered his results in a logically deductive manner, ultimately basing them on a set of self-evident axioms. His number-theoric books introduce the concepts of divisibility, prime numbers, and the greatest common divisor.

The earliest surviving records of the study of prime numbers come from the ancient Greeks. Euclid established fundamental results concerning them, such as the infinitude of primes, the fundamental theorem of arithmetic, and the relation between Mersenne primes and perfect numbers. Another contribution to prime numbers was made by Eratosthenes (3rd century BC). He devised a method to identify all primes up to a given bound. The Sieve of Eratosthenes is still used to construct lists of primes. Euclid gave the Euclidean algorithm for computing the greatest common divisor of two numbers. He presented a formula with two parameters to generate all Pythagorean triples and proved their infinitude.

In his Arithmetica (transl. 1621), Diophantus explored polynomial equations with positive rational variables.

Diophantus (3rd century) was an influential figure in later Greek arithmetic because of his numerous contributions to number theory and his exploration of the application of arithmetic operations to algebraic equations. With his Arithmetica, he systematised polynomial equations with only positive rational variables. His work contains problems involving equations of up to 3rd polynomial degree. For example, the Pythagorean equation is a 2nd degree equation with the Pythagorean triples as its solutions. In modern terms, Diophantine equations contains integer coefficients and solutions, as opposed to positive rational ones.

Chinese motivation for number theory was to solve problems in astronomy and calendar calculations. This included problems that would later be part of modular arithmetic. An anonymous 4th century tretise titled Sunzi Suanjing contains the first example of the Chinese remainder theorem. An exercise seeks for a number that returns the remainders 2, 3, 2 when divided by 3, 5, 7, respectively. The author gave 23 as the solution but there are infinitely many solutions. Althought often cited as the first appearance of the theorem, it is a concrete example and not a general theorem [CN]. The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections.

Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences $n\equiv a_1{modm}_1$, $n\equiv a_2{modm}_2$ could be solved by a method he called kuṭṭaka, or pulveriser; this is a procedure close to (a generalization of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (7th century) started the systematic study of indefinite quadratic equations. He provided a general solution for the linear Diphantine equation $ax+by=c$. He gave solutions to specific cases of Pell's equation. Later authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was found by Jayadeva; the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (12th century).

In the early ninth century, the caliph al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind, which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta). Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī, 953 – c. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.

Thabit ibn Qurra derived a method for generating amicable numbers and found more pairs. He also translated Nicomachus's Introduction to Arithmetic into Arabic. Analogous to amicable numbers, Abu Mansur al-Baghdadi discovered a method to obtain balanced numbers with a given sum. Two integers are called balanced if both sums of the respective proper divisors equal one another. Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) solved problems involving congruences using what is now called Wilson's theorem, characterizing the prime numbers as the numbers $n$ that evenly divide $(n-1)!+1$. In his Opuscula, Alhazen considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci took the innovations from Islamic mathematics to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root.

Pierre de Fermat

Pierre de Fermat (1607–1665) is sometimes considered the founder of modern number theory. His contributions revitalised the study in Western Europe. His work is contained in letters to mathematicians and in private marginal notes like in Diophantus's Arithmetica. Although he drew inspiration from classical sources, in his notes and letters Fermat scarcely wrote any proofs—he had no models in the area.

One of Fermat's first interests was perfect numbers and amicable numbers. These topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day. In 1638, Fermat claimed that all whole numbers can be expressed as the sum of four squares or fewer. Fermat also investigated the primality of the Fermat numbers $2^{2^n}+1$, and Marin Mersenne studied the Mersenne primes, prime numbers of the form $2^p-1$ with $p$ itself a prime. In the field of congruences, he described a fundamental result that would becom Fermat's little theorem: if a is not divisible by a prime p, then $a^{p-1}\equiv 1modp$. It was later proved by Leibniz and Euler. Further, he claimed in Arithmetica that there are no solutions to $x^n+y^n=z^n$ for all $n\ge 3$. In 1657, Fermat posed the problem of solving $x^2-N{y}^2=1$ as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker. Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach, pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following:

Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots$.

• Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that $p=x^2+y^2$ if and only if $p\equiv 1mod4$; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himself); the lack of non-zero integer solutions to $x^4+y^4=z^2$ (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).
• Pell's equation, first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation.
• First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.
• Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form $x^2+N{y}^2$, some of it prefiguring quadratic reciprocity.
• Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.

Carl Friedrich Gauss

Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations; for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to

$m{X}^2+n{Y}^2$

), including defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation

$a{x}^2+b{y}^2+c{z}^2=0$

and worked on quadratic forms along the lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for

$n=5$

(completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain). Carl Friedrich Gauss (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The Disquisitiones Arithmeticae (1801), which he wrote three years earlier when he was 21, had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work the law of quadratic reciprocity and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (congruences) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory:

The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.

In this way, Gauss arguably made forays towards Évariste Galois's work and the area algebraic number theory.

• Erdos's elementary proof of the PNT
• Ramanujan

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