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Friday, 3 March, 2000, 15:19 GMT
The secret of squares revealed
By BBC News Online science editor Dr David Whitehouse
Mathematicians have solved a number mystery that has been puzzling them for hundreds of years. They say it is a major breakthrough in our understanding of numbers and their properties  but it may take a bit of explaining. To mathematicians all numbers are special, but some are more special than others. They say there is deep meaning in numbers so simple that we barely give them a second thought when we use them. Take for instance squares, that is the numbers produced by multiplying numbers by themselves. Using whole numbers you get 1, 4, 9, 16, 25, 36, 49 and so on. Mathematicians call these "perfect squares" and they arise in many fundamental questions about numbers and their properties. For example, Pythagoras's theorem states that a squared + b squared = c squared and is the solution to a rightangled triangle. It is possible to find infinitely many numbers that satisfy this equation. However, the famous Fermat's Last Theorem tells us that if the numbers are not squared but, say, multiplied three times (cubed) then no numbers can be found that will work. Finger counting We count in sets of ten. This seems natural to us because we have ten fingers. However the ancient Babylonians used different units, which is why we measure time in units of 60 minutes and clockfaces have 12 hours. We need not use sets of ten, any number would do. Mathematicians call this modular arithmetic. So we count in modulus ten. When perfect squares and modular arithmetic are combined strange and unexpected things happen. A question mathematicians have wanted to answer for hundreds of year is this: when is a number a perfect square when units are counted in a prime number modulus? (A prime is a special number that can only be divided by 1 and itself, e.g. 2, 3, 5, 7, 11 and so on.) It turns out that the relationship between a number and its square when the counting units are a prime is so surprising that mathematicians have been trying to decide what it means for hundreds of years. Search for proof It was not until the 1960s that mathematician Robert Langlands suggested a reason why the law governing the relationship works  but he could not prove it. Now that proof has been obtained and it is being described as a major milestone in number theory. The result was published in the Notices of the American Mathematical Society. Such research into numbers tells us something profound and deep about the Universe. Even a simple number system has within it complexities that are both subtle and obvious and nobody has yet understood how such number patterns arise out of simple 1 ,2 ,3 . . . 
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