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Fermat's Theorem (stationary Points), Fermat's Theorem
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: * Fermat's Last Theorem, about integer solutions to ''a''''n'' + ''b''''n'' = ''c''''n'' * Fermat's little theorem, a property of prime numbers * Fermat's theorem on sums of two squares, about primes expressible as a sum of squares * Fermat's theorem (stationary points), about local maxima and minima of differentiable functions * Fermat's principle, about the path taken by a ray of light * Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers * Fermat's right triangle theorem, about squares not being expressible as the difference of two fourth powers See also * List of things named after Pierre de Fermat This is a list of things named after Pierre de Fermat, a French amateur mathematician. *Director circle, Fermat–Apollonius circle *Fermat–Catalan conjecture *Fermat cubic *Ferm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Domin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of '' Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Little Theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If is not divisible by , that is if is coprime to , Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: : a^ \equiv 1 \pmod p. For example, if = 2 and = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: :5 = 1^2 + 2^2, \quad 13 = 2^2 + 3^2, \quad 17 = 1^2 + 4^2, \quad 29 = 2^2 + 5^2, \quad 37 = 1^2 + 6^2, \quad 41 = 4^2 + 5^2. On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem (stationary Points)
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. By using Fermat's theorem, the potential extrema of a function \displaystyle f, with derivative \displaystyle f', are found by solving an equation in \displaystyle f'. Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum. Statement One way to state Fermat's theorem is that, if a function has a local extremum at some point and is differentiable there, then the function's derivative at that p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is " stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a ''second-order'' change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in ''very'' close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam. First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig.1), Fermat's principle was initiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Polygonal Number Theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. That is, the -gonal numbers form an additive basis of order . Examples Three such representations of the number 17, for example, are shown below: *17 = 10 + 6 + 1 (''triangular numbers'') *17 = 16 + 1 (''square numbers'') *17 = 12 + 5 (''pentagonal numbers''). History The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared.. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, . Gauss proved the triangular case in 1796, commemorating the occasion by writing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Right Triangle Theorem
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometric forms, it states: *A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square. *A right triangle and a square with equal areas cannot have all sides commensurate with each other. *There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle. More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Things Named After Pierre De Fermat
This is a list of things named after Pierre de Fermat, a French amateur mathematician. *Director circle, Fermat–Apollonius circle *Fermat–Catalan conjecture *Fermat cubic *Fermat curve *Euler's theorem, Fermat–Euler theorem *Fermat number *Fermat point *Weber problem, Fermat–Weber problem *Fermat polygonal number theorem *Diagonal form, Fermat polynomial *Fermat primality test *Fermat pseudoprime *Fermat quintic threefold *Fermat quotient *Difference quotient, Fermat's difference quotient *Fermat's factorization method *Fermat's Last Theorem *Fermat's little theorem *Adequality, Fermat's method *Proof by infinite descent, Fermat's method of descent *Fermat's principle *Fermat's right triangle theorem *Fermat's spiral *Fermat's theorem (stationary points) *Fermat's theorem on sums of two squares *Fermat theory *Pell's equation, Pell–Fermat equation Other *Fermat (computer algebra system) *Fermat (crater) *Fermat Prize {{Pierre de Fermat Lists of things named after ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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