Higher Arithmetic. An Algorithmic Introduction to Number Theory
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Contents:
Geometric number theory traditionally called geometry of numbers incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings.
Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Typical topics include covering system , zero-sum problems , various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field. Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography. Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era.
Does it have rational or integer solutions, and if so, how many? We have to consider how many of the numbers 1 a, 2a, 3a,. For example, it can be proved that the first coefficient a of a reduced form is the least number which is properly represented by the form. For the eighth edition, many people contributed suggestions, notably Dr J. McKean Henry and Moll, V.
The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras , which were written between the 8th and 6th centuries BC. Baudhayana c. Apastamba c. In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE.
The Jaina text Surya Prajinapti c. Each of these was further subdivided into three orders:.
Higher Arithmetic: An Algorithmic Introduction to Number Theory
The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions one dimension , infinite in area two dimensions , infinite everywhere three dimensions , and infinite perpetually infinite number of dimensions. In the Jaina work on the theory of sets , two basic types of transfinite numbers are distinguished.
On both physical and ontological grounds, a distinction was made between asmkhyata and ananata , between rigidly bounded and loosely bounded infinities.
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Number theory was a favorite study among the Hellenistic mathematicians of Alexandria , Egypt from the 3rd century CE, who were aware of the Diophantine equation concept in numerous special cases. The first Hellenistic mathematician to study these equations was Diophantus. Diophantus also looked for a method of finding integer solutions to linear indeterminate equations , equations that lack sufficient information to produce a single discrete set of answers.
Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not. Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations.
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This kuttaka algorithm is considered to be one of the most signicant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy.
He also found the general solution to the indeterminate linear equation using this method. Brahmagupta in handled more difficult Diophantine equations. His Brahma Sphuta Siddhanta was translated into Arabic in and was subsequently translated into Latin in The general solution to this particular form of Pell's equation was found over 70 years later by Euler , while the general solution to Pell's equation was found over years later by Lagrange in Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in , using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations.
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Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over years later. Bhaskara also found solutions to other indeterminate quadratic, cubic , quartic and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations. From the 9th century, Islamic mathematicians had a keen interest in number theory. The first of these mathematicians was the Arab mathematician Thabit ibn Qurra , who discovered a theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other.
To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page. MathWorld Book. Topics in a Number Theory Course To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page. A congruence is an equation in modular arithmetic, i. Continued Fraction. A continued fraction is a real number expressed as a nested fraction. Such representations may be particualrly useful in number theory. Diophantine Equation. Divisor Function.
The divisor function of order k is the number theoretic function that gives the sum of k th powers of divisors of a given integer. Elliptic Curve. Euclidean Algorithm. Many good problems. Has a wonderful collection of problems. Highly recommended. Bourbaki does everything in the utmost generality. Everything in this book is done with a mind to the student. Out of print, but can be bought online.
Excellent for learning. Moves slowly and gives examples, but it can be hard to see the forest for the trees.
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Nota Bene: Many of the journal articles in the subject are accessible. The new, expanded second edition is called Topology. Bypasses Lie algebra theory. Requires college-level mathematical maturity. Contains many exercises, as notorious as they are pedagogically sound. Highly recommended as a first book on manifolds.
Number theory
Perfect for those comfortable with a fairly high level of abstraction. Elementary and very readable. Teaches in Scheme, a dialect of Lisp. Corman, Leiserson, Rivest, and Stein: Algorithms 2nd ed.