Advances in Number Theory between 1900 and 1949

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Advances in Number Theory between 1900 and 1949

Overview

The latter part of the nineteenth century and the first half of the twentieth saw major advances in many branches of mathematics, including the theory of numbers. Work on classic problems in the field led to important advances in our understanding of numbers and their relationships to each other. New work by both renowned and emerging mathematicians pointed the way to solutions of other classic problems, and also promised future advances. Much of this work focused on prime numbers, numbers that are only divisible by 1 and themselves. Although prime numbers have enchanted mathematicians for centuries, only in the last century or so did many of their properties begin to be better understood, including how to generate them and how to factor very large numbers into their prime components. Recent improvements in computer technology has made prime numbers central to many methods of encrypting information for personal, business, or governmental security.

Background

A number is an abstraction. It is difficult to explain to someone what, for example, the number 5 is without directly or indirectly referring to the number 5 itself. However, there are few arguments in daily life about the properties of the number 5. Once we learn to count, we all agree that 5 is greater than 4 and less than 6. After studying a little mathematics in school, we also agree that 5 is a prime number, one that is evenly divisible only by itself and the number 1. Further study may reveal other properties of this number, such as, when multiplied by even numbers, the product invariably ends in a 0 while, when multiplied by odd numbers, the product ends in a 5.

If we extend our inquiries a bit further, we may also notice that there are other prime numbers in the "vicinity" of 5; 2 and 3 as well as 7 and 11. We might also notice that prime numbers are not evenly grouped (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc.), that (with the exception of 2) they are all odd numbers, and as numbers grow larger, primes generally become fewer and farther between. There are many other ways that numbers form groups, patterns, or have certain properties. The field of mathematical inquiry that studies these patterns and properties of numbers is called number theory.

Number theory is a relatively new subdiscipline of mathematics, which was first recognized in the nineteenth century as a formal field of study. Earlier mathematicians had, of course, done work in the field, including, most famously, Pierre de Fermat (1601-1665), Edward Waring (1734?-1798), Christian Goldbach (1690-1764), and others. In the first half of the twentieth century, equally brilliant mathematicians, including David Hilbert (1862-1943), Ivan Vinogradov (1891-1983), Srinivasa Ramanujan (1887-1920), Godfrey Hardy (1877-1947), and John Littlewood (1885-1977), made great advances in some areas of number theory, in some cases finding solutions to problems originally posed a century or more earlier.

In 1742, Christian Goldbach proposed that all even numbers larger than 4 were the sum of two odd prime numbers. Goldbach's conjecture, as this came to be called, has been proven numerically (that is, by computation) for a great many numbers, up to very large values, but the general case (i.e., for all numbers) has remained unproven to this day. A partial solution to Goldbach's conjecture, however, was proven in 1937 by the Russian Vinogradov, when he showed that every sufficiently large odd number is the sum of three odd prime numbers. This is, admittedly, a far cry from Goldbach's conjecture because it is less general. However, it was much closer to a solution than had been previously achieved and, perhaps, it will inspire mathematicians in the direction of a true general solution to this problem.

Another eighteenth-century mathematician, Edward Waring, proposed in 1770 that any number is the sum of no more than 9 cubes (a number multiplied by itself three times) or 19 fourth powers. Unfortunately, Waring proposed this without mathematical proof and he stopped at fourth powers, without extending his proposition any further. Waring's problem, as it came to be called, occupied mathematicians because it was something that seemed as though it should be true for all numbers, but that seemingly defied either proof or extension into higher powers.

In 1909 the great German mathematician David Hilbert was able to find a solution to Waring's problem, making a number of significant advances in number theory while doing so. Hilbert extended the boundaries of the problem by finding a general solution, valid for any number and for any power of equation.

Other important work in number theory was undertaken by English mathematician Godfrey Hardy, in collaboration with both John Littlewood and Srinivasa Ramanujan. With Littlewood, Hardy published a several important papers on number theory in which they examined the distribution of prime numbers (as discussed very briefly above) to see if any patterns could be discerned. For example, prime numbers become more scarce as numbers increase, but is there any way to determine how many primes exist between, say, 0 and 1,000 compared with the interval 9,000 to 10,000? Or, is there any way to predict the occurrence of "twin" primes; prime numbers pairs like 41 and 43 that are consecutive odd numbers? And, for that matter, are there any formulae that can be used to reliably generate prime numbers? For example, the formula x² + x + 41 generates 40 consecutive prime numbers, and then it stops being nearly as effective.

Harding's other major collaboration was with the Indian mathematical prodigy Srinivasa Ramanujan, a largely self-taught mathematician whose pursuit of mathematics at the expense of all his other studies led to his dismissal from the University of Madras in 1903. Following a year's correspondence and several important publications, Hardy arranged for Ramanujan to travel to England for joint work and tutoring. Their work, in conjunction with Ramanujan's own efforts, led to significant advances in a variety of areas, including number theory. In some areas, however, his lack of formal mathematics education showed, particularly in some aspects of prime number theory. However, these errors were not important compared to his mastery of other aspects of mathematics and, upon his death from tuberculosis at the age of 33, was generally regarded by his fellow mathematicians as one of those true mathematical geniuses who only grace us every century or so.

Impact

Many of the advances in number theory outlined above have little applicability to what most of us call "real life." Some, like the proof of Fermat's Last Theorem, arouse public interest for a short time, and then fade into obscurity. Others fail to create even a minor stir in the general public. From that perspective, advances that are of utmost importance and interest to mathematicians are often of no import to the nonmathematical world because nonmathematicians often fail to see any practical impact of these discoveries on their lives.

This attitude, while understandable, may result from the perception that mathematicians want to keep their field aloof from the rest of the world or from any practical application. While this undoubtedly has some justification, it is simply not correct. Even if there is no direct application, for example, of the Goldbach conjecture on daily life, it should be of at least passing interest because it tells us something about our world. We all use numbers daily, in counting change, picking flowers, numbering the pages of a book, and so on. We may not all sit and contemplate the fact that 2, 3, and 5 are the only prime numbers that close to one another, but knowing this and other facts about the numbers that crop up so frequently in our lives enriches us in at least a small way. In addition, there is at least one very important and very practical way that research into prime numbers that affects many people: the use of large prime numbers in encrypting data.

Prime number theory, although of interest to mathematicians, was of little practical utility until recently. In the 1990s, however, with the advent of very fast and inexpensive desktop computers, prime numbers became very important in the field of encryption and data security. In fact, the ability to find, multiply, and divide by very large prime numbers is the basis for generating secure military, commercial, and private encryption systems. Any algorithm that makes generating large prime numbers faster and easier aids encryption efforts while any algorithm that makes factoring very large numbers faster and easier aids those trying to break such codes. The most secure system currently available to the public (at the time this was written) uses very large prime numbers (larger than 3 × 10³⁸) to encrypt messages. The only way to decrypt the message is to factor this large number into its component primes and use them to restore the message to its original form. Obviously, any research in number theory that helps to generate or factor such large primes is relevant to cryptography, making data either more or less secure and making governments either more or less assured.

P. ANDREW KARAM