List of mathematical contexts in which exponentiated terms are summed
In mathematics and statistics, sums of powers occur in a number of contexts:
Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
There are only finitely many positive integers that are not sums of distinct squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See [1] for a generalization to sums of polynomials.
Faulhaber's formula expresses as a polynomial in n, or alternatively in terms of a Bernoulli polynomial.
Fermat's right triangle theorem states that there is no solution in positive integers for and .
Fermat's Last Theorem states that is impossible in positive integers with k > 2.
The equation of a superellipse is . The squircle is the case k = 4, a = b.
Euler's sum of powers conjecture (disproved) concerns situations in which the sum of n integers, each a kth power of an integer, equals another kth power.
The Fermat-Catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1.
Beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2.
The Jacobi–Madden equation is in integers.
The Prouhet–Tarry–Escott problem considers sums of two sets of kth powers of integers that are equal for multiple values of k.
A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in n distinct ways.
The Riemann zeta function is the sum of the reciprocals of the positive integers each raised to the power s, where s is a complex number whose real part is greater than 1.
The Lander, Parkin, and Selfridge conjecture concerns the minimal value of m + n in
Waring's problem asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most skth powers of natural numbers.
The successive powers of the golden ratio φ obey the Fibonacci recurrence:
Newton's identities express the sum of the kth powers of all the roots of a polynomial in terms of the coefficients in the polynomial.
The sum of cubes of numbers in arithmetic progression is sometimes another cube.
The Fermat cubic, in which the sum of three cubes equals another cube, has a general solution.
The power sum symmetric polynomial is a building block for symmetric polynomials.
The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1.
The Erdős–Moser equation, where m and k are positive integers, is conjectured to have no solutions other than 11 + 21 = 31.
The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown whether all remaining integers can be expressed in this form.
The sum of the terms in the geometric series is
^Graham, R. L. (June 1964). "Complete sequences of polynomial values". Duke Mathematical Journal. 31 (2): 275–285. doi:10.1215/S0012-7094-64-03126-6. ISSN 0012-7094.
In mathematics and statistics, sumsofpowers occur in a number of contexts: Sumsof squares arise in many contexts. For example, in geometry, the Pythagorean...
that cannot be expressed as a sumof three cubes? (more unsolved problems in mathematics) In the mathematics ofsumsofpowers, it is an open problem to characterize...
elsewhere, sumsof squares occur in a number of contexts: For partitioning of variance, see Partition ofsumsof squares For the "sumof squared deviations"...
list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here...
for sumsofpowers using symbolic notation, but even he calculated only up to the sumof the fourth powers. Johann Faulhaber gave formulas for sumsof powers...
any length, however great. As of 2020[update] the conjecture remains unproven. Large set Sumof squares Sumsofpowers Unless given here, references are...
provides a solution to the Prouhet–Tarry–Escott problem of finding sets of numbers whose sumsofpowers are equal up to the k {\displaystyle k} th power. In...
works included a 1949 paper on sumsofpowers, showing that almost all positive integers could be represented as a sumof a square, a cube, and a fourth...
expansion ofpowersof a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc...
be represented as sumsof consecutive positive integers are called polite numbers; they are exactly the numbers that are not powersof two. The geometric...
progressions Utonality Polynomials calculating sumsofpowersof arithmetic progressions Hayes, Brian (2006). "Gauss's Day of Reckoning". American Scientist. 94 (3):...
where the sumof the squares of three equals the square of the fourth Sumsof three cubes – Problem in number theory Sumsofpowers – List of mathematical...
only the sign of a number, as +1 or −1. Absolute value: distance to the origin (zero point) Sigma function: Sumsofpowersof divisors of a given natural...
Sumer (/ˈsuːmər/) is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during...
solved the open problem posed by Davenport on writing numbers as the sumsof fifth powers, Conway began to become interested in infinite ordinals. It appears...
list of fictional characters from the Austin Powers series of films. Austin Powers is a series of American spy action comedy films: Austin Powers: International...
arithmetic was used to disprove Euler's sumofpowers conjecture on a Sinclair QL microcomputer using just one-fourth of the integer precision used by a CDC...
known that there are an infinite number of such sums involving coprime 3-powerful numbers; however, such sums are rare. The smallest two examples are:...