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Applied number theory

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  Applied number theory Number theory used to be considered the purest of pure math. Leonard Dickson once said “Thank God that number theory is unsullied by any application.” Dickson died in 1954. Had he lived a little longer he would not have said what he did. There were applications of number theory in Dickson’s day, but many have been developed more recently. The best known application of number theory is public key cryptography, such as the RSA algorithm. Public key cryptography in turn enables many technologies we take for granted, such as the ability to make secure online transactions. In addition to cryptography, number theory has been applied to other areas, such as: Error correcting codes Numerical integration Computer arithmetic Random and quasi-random number generation
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Probabilistic number theory

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  Probabilistic number theory [ edit ] Main article:  Probabilistic number theory Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually  independent . For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite. It is sometimes said that  probabilistic combinatorics  uses the fact that whatever happens with probability greater than  0  must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement foll...
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Diophantus

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  Diophantus [ edit ] Title page of the 1621 edition of  Diophantus 's  Arithmetica , translated into  Latin  by  Claude Gaspard Bachet de Méziriac . Very little is known about Diophantus of Alexandria; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's  Arithmetica  survive in the original Greek and four more survive in an Arabic translation. The  Arithmetica  is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form  � ( � , � ) = � 2  or  � ( � , � , � ) = � 2 . Thus, nowadays, we speak of  Diophantine equations  when we speak of polynomial equations to which rational or integer solutions must be found. One may say that Diophantus was studying  rational points , that is, points whose coordinates are rational—on  curves ...
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