Fermat's Last Theorem (Conditions and decisions)
1. Pierre de Fermat
Pierre Fermat
The French lawyer and mathematician Pierre de Fermat or Petri de Fermat (1601–1665), with his work had a great influence on the further development of mathematics.
Pierre Fermat is one of the founders of number theory. He took this important step in his works on the largest and smallest quantities, which opened a series of studies by Fermat, which is one of the largest links in the history of the development of not only higher analysis in general, but also the analysis of infinitesimals in particular.
Fermat made the following statement: if a number a is not divisible by a prime number p, then there is an exponent k such that a-1 is divisible by p, and k is a divisor of p-1. This statement is called Fermat's little theorem. It is fundamental in all elementary number theory.
Pierre Fermat, according to the rules accepted in mathematics of the 19th-21st centuries, found tangents to algebraic curves.
In mathematical analysis, Fermat's lemma or the necessary criterion for an extremum is used: at extremum points, the derivative of a function is equal to zero.
Fermat developed a method for systematically finding all divisors of a number and formulated a theorem on the possibility of representing an arbitrary number by a sum of no more than four squares.
In the field of the infinitesimal method, he systematically studied the process of differentiation, gave a general law for differentiation of powers, and applied this law to the differentiation of fractional powers. In the preparation of modern methods of differential calculus, his creation of the rule for finding extrema was of great importance.
P. Fermat formulated the general law of differentiation of fractional powers and extended the formula for integrating powers to the cases of fractional and negative exponents.
Pierre Fermat developed the foundations of probability theory. In Fermat's works, both basic processes of the infinitesimal method were systematically developed, but he ignored the connection between the operations of differentiation and integration. This connection was made later.
Fermat was the first to come up with the idea of coordinates and created analytical geometry, introducing an infinitesimal quantity. He solved the problem of squaring any curve, and on this basis solved a number of problems on finding the centers of gravity. In the work “Introduction to the Theory of Plane and Spatial Places,” he was the first to classify curves depending on the order of their equation, establishing that a first-order equation defines a straight line, and a second-order equation defines a conic section. Developing these ideas, he applied analytical geometry to space.
In the field of physics, Fermat is associated with the establishment of the basic principle of geometric optics, by virtue of which light in an inhomogeneous medium chooses the path that takes the least time (Fermat believed that the speed of light is infinite and formulated the principle more vaguely). With this thesis begins the history of the main law of physics - the principle of least action.
The first collected works of P. Fermat, “Various Works,” was published in 1679.
2. The problem of dividing a square into the sum of two squares
P. Fermat often wrote down not proofs, but only brief instructions about the method he used.
