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Calculus is the most useful mathematical tool of all, and scholars eagerly applied its ideas to a range of real-world problems. Euler was the most prolific mathematician ever—and one of the most influential—and when he turned his attention to number theory, the subject could no longer be ignored.

Like an insistent salesman, Goldbach tried to interest Euler in the theory of numbers, and eventually his insistence paid off.

### Basic method

He later took up the matter of perfect numbers, demonstrating that any even perfect number must assume the form discovered by Euclid 20 centuries earlier see above. Of course, even Euler could not solve every problem.

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Euler endorsed the result—today known as the Goldbach conjecture —but acknowledged his inability to prove it. Euler gave number theory a mathematical legitimacy, and thereafter progress was rapid.

## Fermat's Last Theorem by Simon Singh – book review

Of immense significance was the publication of Disquisitiones Arithmeticae by Carl Friedrich Gauss — This became, in a sense, the holy writ of number theory. In it Gauss organized and summarized much of the work of his predecessors before moving boldly to the frontier of research.

He also gave the first proof of the law of quadratic reciprocity , a deep result previously glimpsed by Euler. To expedite his work, Gauss introduced the idea of congruence among numbers—i. Inspired by Gauss, other 19th-century mathematicians took up the challenge.

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This surprising but ingenious strategy marked the beginning of a new branch of the subject: analytic number theory. Number theory. Info Print Print. Table Of Contents.

## Fermat's Last Theorem - Wikipedia

Submit Feedback. Thank you for your feedback. Introduction From prehistory through Classical Greece Pythagoras Euclid Diophantus Number theory in the East Modern number theory Pierre de Fermat Number theory in the 18th century Number theory in the 19th century Disquisitiones Arithmeticae From classical to analytic number theory Prime number theorem Number theory in the 20th century Unsolved problems.

Load Previous Page. Pierre de Fermat Credit for changing this perception goes to Pierre de Fermat —65 , a French magistrate with time on his hands and a passion for numbers.

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Read More on This Topic. This theorem is one of the great tools of modern number theory.

## Number theory

This has finally been proven by Wiles in Here we are concerned with his "little" but perhaps his most used theorem which he stated in a letter to Fre'nicle on 18 October Fermat's Little Theorem. It is so easy to calculate a p -1 that most elementary primality tests are built using a version of Fermat's Little Theorem rather than Wilson's Theorem. As usual Fermat did not provide a proof this time saying "I would send you the demonstration, if I did not fear its being too long" [ Burton80 , p79].

Euler first published a proof in , but Leibniz left virtually the same proof in an unpublished manuscript from sometime before Start by listing the first p -1 positive multiples of a : a , 2 a , 3 a ,