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Pierre de Fermat , born the August 20th 1601 or in 1607, with Beaumont-with-Lomagne, close to Montauban, and dead the January 12th 1665 with Castrate, is a lawyer and mathematician French, called “the prince of the amateurs”.

Biography

His/her father, Dominique Fermat, were an easy, middle-class merchant and second Consul of the city as leather merchant and other food products. One does not know where it carried out its primary studies. Thereafter, it makes studies with Toulouse and of Droit to Orleans. As of 1631, it buys a responsibility of advising of the king to the Room of the Requests of the Parlement of Toulouse. He marries this year Louise de Long with which he will have five children. With fidelity and insurance in this use of magistrate, it fills his task and quickly climbs the levels towards an use of notable with the Court of criminal appeal and the Grand' Room and finally, member of the room of the edict of Castres (1648). It is at the latter station that a particle of nobility is added to its name and it names henceforth Pierre de Fermat.

Its talents of mathematician were exerted with share of its work of magistrate since the great writings which one found of him are annotations in famous texts the such Arithmetica of Diophante and part of his correspondence with the scientists of the 17th century. Its formation as a mathematician is only little known: it seems that he studied works of François Viète which he finds in the library of a friend, Etienne d' Espagnet.

His friends mathematicians (Descartes, Pascal, Roberval, Torricelli, Huygens, Mersenne), he asks to show by the proof the theories which he advances what revives the anger of the others towards him. He disputes in particular with Descartes in 1637. In 1652, the famous plague which devastates France will attack with him but it will face there and will fight it. It is only in 1670 that its theorem is exposed to the public. He comments on, by extending it, Diophante, and restores with an admirable sagacity several lost works of Apollonius ( De Locis planis , plane places, in 1636) and of Euclide. He is at the same time skilful a hellenist and deep a Jurisconsulte. This scientist hid his methods, of which some were lost with him.

It was also interested in physical sciences; one owes him in particular the Principe of Fermat in optics.

After its death

It remains after its death only one important correspondence dispersed in all Europe.

The son of Pierre de Fermat publishes, in 1670, an edition of the Arithmetica of Diophante, annotated by his father, then in 1679 a series of articles and a selection of its correspondence under the name of Varia opera mathematica .

In 1839, Guglielmo Libri tries to withdraw a certain number of manuscripts, of which a part only will be recovered.

Charles Henry and Paul Tannery publish, at the beginning of the XXe century, the Œuvres of Fermat in four volumes and a supplement (1922).

Contributions

He shares with Descartes glory to have applied the algebra to the geometry. He imagined for the solution of the problems, a method, known as of maximis and minimis, which makes it look like the first inventor of the differential Calculus of which he is a precursor: he is the first to use the formula (if not the concept) of the derived number.

He poses at the same time as Blaise Pascal the bases of the probability theory. But its major contribution relates to the Théorie of the numbers and the equations diophantiennes. Author of several theorems or conjectures in this field, it is in the middle of the “modern theory of the numbers”.

It is very known for two “theorems”:

• the “Small theorem of Fermat”;
• the “Last theorem of Fermat”; this last was only a Conjecture and remained during nearly three centuries of feverish research.

Small theorem of Fermat

If p is a Prime number and has a Entier nondivisible naturalness by p , then $a^\; \{p-1\}\; \backslash \; equiv\; 1\; \backslash \; pmod\; \{p\}$.

See also: Theorem of Euler , whose this theorem is a particular case.

Theorem of the two squares of Fermat

So that a prime number odd is the sum of two squares, it is necessary and it is enough that it is adequate to 1 modulo 4.

Theorem of Fermat on the polygonal numbers

Entire is written:

• like summons with more the 3 triangular numbers
• like summons with more the 4 square numbers
• like summons with more the 5 pentagonal numbers
triangular
• etc
• numbers : $1;\; 3\; (=1+2);\; 6\; (=1+2+3);\; 10\; (=1+2+3+4)\dots \; \backslash !$: the nth triangular number is equal to the sum of N first whole natural nonnull;
• square numbers : $1;\; 4\; (=1+3);\; 9\; (=1+3+5);\; 16\; (=1+3+5+7)\dots \; \backslash ,\; \backslash !$: the nth square number is equal to the sum of N first odd natural entireties)
• pentagonal numbers : $1;\; 5\; (=1+4);\; 12\; (=1+4+7);\; 22\; (=1+4+7+10)\dots \; \backslash ,\; \backslash !$: the nth pentagonal number is equal to the sum of N first natural entireties adequate with 1 modulo 3;
• polygonal numbers of order m : $1;\; 1+\; (M-1);\; 1+\; (M-1)\; +\; (2m-3);\; 1+\; (M-1)\; +\; (2m-3)\; +\; (3m-5);\; \dots \; \backslash ,\; \backslash !$: the nth polygonal number of order m is equal to the sum of N first adequate natural entireties with 1 modulo m2.

This theorem was stated by Fermat, was shown in the case of the square numbers by Jacobi and, independently by Joseph-Louis Lagrange at the 18th century (This last making use of partial results obtained by Euler). Gauss solved the case of the triangular numbers in 1796. A complete proof was proposed by Cauchy in 1813.

Great theorem of Fermat (or Last theorem of Fermat)

There do not exist overall strictly positive entireties $x\; \backslash ,\; \backslash !$, $y\; \backslash ,\; \backslash !$, $z\; \backslash ,\; \backslash !$ checking the equation $x^n\; +\; y^n\; =\; z^n\; \backslash ,\; \backslash !$ when $n$ is an entirety such as $n\; >\; 2\; \backslash ,\; \backslash !$.

This theorem was shown by the English mathematician Andrew Wiles of the University of Princeton, with the assistance of Richard Taylor. After a first presentation in June 1993, then the discovery of an error and a year of additional work, the proof was finally published in 1995 in Annals off Mathematics .

Pierre de Fermat itself annotated in margin of his specimen of the Arithmetic ones that he had discovered a really remarkable demonstration of it, but lacked place to give it to this place: " I discovered a really remarkable proof that this too narrow margin does not allow me a détailler".

The demonstration evoked by Pierre de Fermat is either distorts, or unknown to date, because the demonstration carried out by Andrew Wiles uses mathematical tools whose Mr. de Fermat could not probably have taking into account knowledge his time.

Method of the infinite descent

Fermat is in addition the inventor of a method of demonstration, the infinite Descente. It consists in showing that if a proposal P is true with a row R , it is it with a row Q lower than R . If one leads to a contradiction, one shows whereas P is false. This very astute method was used by Fermat to show its great theorem in the particular case N = 4 .

Principle of Fermat (optical)

The way traversed by the light between two points is always that which minimizes run time. See the article Principle of Fermat

References

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