Concept
Lagrange's four-square theorem
Related concepts (13)
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: with x and y integers, if and only if 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: 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.
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (dʒəˈkoʊbi; jaˈkoːbi; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasionally written as Carolus Gustavus Iacobus Iacobi in his Latin books, and his first name is sometimes given as Karl. Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of banker Simon Jacobi.
Sum of two squares theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer > 1 to whether it can be written as a sum of two squares, such that = ^2 + ^2 for some integers a, b. An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor ^k, where prime and k is odd. In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way.
Pierre de Fermat
Pierre de Fermat (pjɛʁ də fɛʁma; 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.
Fermat polygonal number theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-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 n-gonal numbers form an additive basis of order n. 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).
Waring's problem
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909.
Adrien-Marie Legendre
Adrien-Marie Legendre (ləˈʒɑːndər,_-ˈʒɑːnd; adʁiɛ̃ maʁi ləʒɑ̃dʁ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770.
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
Euclidean algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use.
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form where a, b, c, and d are real numbers; and 1, i, j, and k are the basis vectors or basis elements.