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For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the 1930s and 1940s, and computational complexity theory from the 1970s.
- Overview
- From prehistory through Classical Greece
- Pythagoras
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.
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.
Britannica Quiz
Numbers and Mathematics
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.
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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.)
2 days ago · Many of the oldest questions in number theory involve what are now known as Diophantine equations: polynomial equations in multiple variables with integer coefficients, where the unknowns are constrained to be integers as well. Indeed, the problem of finding the solutions to the simplest linear Diophantine equation, \(ax+by=c,\) is essentially ...
Oct 17, 2011 · 1 Early Roots to Fermat. Number theory, the study of the properties of the positive integers, which broadened in the 19th century to include other types of “integers,” is one of the oldest branches of mathematics. It has fascinated both amateurs and mathematicians throughout the ages.
- Israel Kleiner
- kleiner@rogers.com
- 2012
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Still other number theory conjectures, both old and new, remain unproofed. Numbers are as infinite as human understanding is finite, so number theory and its various subfields will continue to captivate the minds of math lovers for ages. Old problems may fall, but new and more complicated conjectures will rise.
Number theory—the study of properties of the positive integers—is one of the oldest branches of mathematics. It has fascinated both amateurs and mathematicians throughout the ages. The subject is tangible, and a great many of its problems are simple to state yet very difficult to solve.
So what is "arithmetic," or number theory? Simply stated, number theory is concerned with questions about and properties of the integers \[\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots onumber \] and closely-related numbers.
