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In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.
Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations ‘sin’, ‘cos’ and ‘tan’ for the trigonometric functions.
Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)
Introduction
We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.
More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.
Continue reading ‘Waring’s Problem & Lagrange’s Four-Square Theorem’
God may not play dice with the Universe, but something strange is going on with the prime numbers [Paul Erdös, paraphrasing Albert Einstein]
The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’
This week, That’s Maths in the Irish Times ( TM022 ) reports on two exciting recent breakthroughs in prime number theory.
The mathematics we study at school gives the impression that all the big questions have been answered: most of what we learn has been known for centuries, and new developments are nowhere in evidence. In fact, research in maths has never been more intensive and advances are made on a regular basis.
The theme of That’s Maths (TM008) this week is prime numbers. Almost all the largest primes found in recent years are of a particular form M(n) = 2n−1. They are called Mersenne primes. The Great Internet Mersenne Prime Search (GIMPS) is aimed at finding ever more prime numbers of this form. Continue reading ‘A Mersennery Quest’
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