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Number Theory

  Number theory, in mathematics, is the study of numbers, and, in particular, of the relationship between the operations of addition and multiplication. Number theory goes right back to the beginning of mathematics, but was hindered from growth by the lack of a clear system of notation for numbers until the end of the Middle Ages. It is an area in which the problems can be very easily stated and understood, but where there are still many unsolved basic questions. After thousands of years of study, the relationship between addition and multiplication remains one of the most mysterious in the whole of mathematics.

One of the key concepts in number theory is that of the prime. A prime number is one which has no divisors (numbers that divide into it exactly, leaving no remainder); the smallest examples are 2, 3, 5, 7, 11, 13 and 17. The ancient Greeks knew that there were infinitely many primes (the reasoning for this is that if there were infinitely many, then you could multiply them all together and then add 1 to the product, producing a number which none of the primes divide; this number must therefore be prime, which is a contradiction; so that there must be infinitely many), but many other properties remain unproven. For example, are there infinitely many pairs of primes that differ only by 2, like 11 and 13? Another famous example is the Goldbach conjecture, which has been checked by computer for vast numbers, but which remains unproved, which is that every even number (greater than 2) is the sum of two primes. There are many such conjectures.

One of the most famous number theoretic conjectures is known as Fermat\'s last theorem. There are numbers x, y and z such that x2 + y2 = z2 (x = 3, y = 4, z = 5, for example), but are there any numbers such that xn + yn = zn, for any n greater than 2? The conjecture is that there are none, so that 2 is unique in this respect. The reason for the name (which would usually indicate that Pierre de Fermat (1601 - 1665) proved the result) is that in one of the books in Fermat\'s library, there was a marginal note scribbled to the effect that he had discovered a proof, but it was too long to fit in the margin—a most frustrating state of affairs. (Mathematical journals still receive new ‘proofs’ of Fermat\'s theorem every week, but they are all found to be wrong.)

Another major area in number theory is modular arithmetic. This involves doing arithmetic in the normal way, but only looking at the remainders upon division by some fixed number, known as the base or modulus. It is like doing arithmetic in the number system based on the modulus and then only looking at the units column in the answer to sums. When looked at in this way, the numbers become far simpler, and properties can be seen which were hidden by all the extra structure there before. For any modulus n, the numbers modulo n form a ring, and if n is a prime, this ring is in fact a field, so that the powerful weapons of abstract algebra can be brought to play against the various problems. SMcL



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