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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. * Fermat–Apollonius circle *Fermat–Catalan conjecture *Fermat cubic *Fermat curve * Ferma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; ; 17 August 1601 – 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 1601 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, Dominique Fermat, was a wealthy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. 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 mainstream 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 (ZFC), or of a less powerful theory, such as Peano arithmetic. 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 ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...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 number, positive integers , , and satisfy the equation for any integer value of greater than . 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, Wiles's proof of Fermat's Last Theorem, the first success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Little Theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , then , and is an integer multiple of . If is not divisible by , that is, if is coprime to , then 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 and , then , and is a multiple of . 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 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Pierre de Fermat, Fermat's theorem on sums of two squares states that an Even and odd numbers, odd prime number, 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 Modular_arithmetic#Congruence, congruent to 1 modular arithmetic, 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 (if number squared is even) or 1 (if number squared is odd) modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem (stationary Points)
In mathematics, the interior extremum theorem, also known as Fermat's theorem, is a theorem which states that at the local extrema of a differentiable function, its derivative is always zero. It belongs to the mathematical field of real analysis and is named after French mathematician Pierre de Fermat. By using the interior extremum theorem, the potential extrema of a function f, with derivative f', can found by solving an equation involving f'. The interior extremum 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. History Pierre de Fermat proposed in a collection of treatises titled ''Maxima et minima'' a method to find maximum or minimum, similar to the modern interior extremum theorem, albeit with the use of infinitesimals rather than derivatives ... [...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 geometrical optics, ray optics and physical optics, wave optics. Fermat's principle states that the path taken by a Ray (optics), ray between two given points is the path that can be traveled in the least time. First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the Snell's law, ordinary law of refraction of light (Fig.1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. If points ''A'' and ''B'' are given, a wavefront expanding from ''A'' sweeps all possible ray paths radiating from ''A'', whether they pass through ''B'' or not. If the wavefront reaches point ''B'', it sweeps not only the ''ray'' path(s) from ''A'' to ''B'', but also an infinitude of near ... [...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 writi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Right Triangle Theorem
Fermat's right triangle theorem is a non-existence mathematical proof, 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 many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometry, 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 (algebra), 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 Commensurability (mathematics), commensurate with each other. *There do not exist two Pythagorean triple, 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 equa ... [...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. * Fermat–Apollonius circle *Fermat–Catalan conjecture *Fermat cubic *Fermat curve * Fermat–Euler theorem *Fermat number * Fermat point * Fermat–Weber problem *Fermat polygonal number theorem * Fermat polynomial * Fermat primality test * Fermat pseudoprime * Fermat quintic threefold *Fermat quotient * Fermat's difference quotient * Fermat's factorization method *Fermat's Last Theorem *Fermat's little theorem * Fermat's method * 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–Fermat equation * 12007 Fermat Other * Fermat (computer algebra system) * Fermat (crater) *Fermat Prize {{DEFAULTSORT:Things named after Pierre de Fermat Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
