# Sums of powers

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{{Short description|List of mathematical contexts in which exponentiated terms are summed}}
In [mathematics](/source/mathematics) and [statistics](/source/statistics), '''sums of powers''' occur in a number of contexts:

*[Sums of squares](/source/Sum_of_squares_(disambiguation)) arise in many contexts. For example, in [geometry](/source/geometry), the [Pythagorean theorem](/source/Pythagorean_theorem) involves the sum of two squares; in [number theory](/source/number_theory), there are [Legendre's three-square theorem](/source/Legendre's_three-square_theorem) and [Jacobi's four-square theorem](/source/Jacobi's_four-square_theorem); and in [statistics](/source/statistics), the [analysis of variance](/source/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 <ref>{{Cite journal |last=Graham |first=R. L. |author-link=Ronald Graham |date=June 1964 |title=Complete sequences of polynomial values |url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-31/issue-2/Complete-sequences-of-polynomial-values/10.1215/S0012-7094-64-03126-6.full |journal=Duke Mathematical Journal |volume=31 |issue=2 |pages=275–285 |doi=10.1215/S0012-7094-64-03126-6 |issn=0012-7094|url-access=subscription }}</ref> for a generalization to sums of polynomials.
*[Faulhaber's formula](/source/Faulhaber's_formula) expresses <math>1^k + 2^k + 3^k + \cdots + n^k</math> as a polynomial in {{mvar|n}}, or alternatively [in terms of a Bernoulli polynomial](/source/Bernoulli_polynomials).
*[Fermat's right triangle theorem](/source/Fermat's_right_triangle_theorem) states that there is no solution in positive integers for <math>a^2=b^4+c^4</math> and <math>a^4=b^4+c^2</math>.
*[Fermat's Last Theorem](/source/Fermat's_Last_Theorem) states that <math>x^k+y^k=z^k</math> is impossible in positive integers with {{math|''k'' > 2}}.
*The equation of a [superellipse](/source/superellipse) is <math>|x/a|^k+|y/b|^k=1</math>. The [squircle](/source/squircle) is the case  {{math|1=''k'' = 4}}, {{math|1=''a'' = ''b''}}.
*[Euler's sum of powers conjecture](/source/Euler's_sum_of_powers_conjecture) (disproved) concerns situations in which the sum of {{mvar|n}} integers, each a {{mvar|k}}<sup>th</sup> power of an integer, equals another {{mvar|k}}<sup>th</sup> power.
*The [Fermat-Catalan conjecture](/source/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](/source/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](/source/Jacobi%E2%80%93Madden_equation) is <math>a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4 </math> in integers.
*The [Prouhet–Tarry–Escott problem](/source/Prouhet%E2%80%93Tarry%E2%80%93Escott_problem) considers sums of two sets of {{mvar|k}}<sup>th</sup> powers of integers that are equal for multiple values of {{mvar|k}}.
*A [taxicab number](/source/taxicab_number) is the smallest integer that can be expressed as a sum of two positive third powers in {{mvar|n}} distinct ways.
*The [Riemann zeta function](/source/Riemann_zeta_function) is the [sum of reciprocals](/source/sum_of_reciprocals) of the positive integers each raised to the power {{mvar|s}}, where {{mvar|s}} is a [complex number](/source/complex_number) whose real part is greater than 1.
*The [Lander, Parkin, and Selfridge conjecture](/source/Lander%2C_Parkin%2C_and_Selfridge_conjecture) concerns the minimal value of {{math|''m'' + ''n''}} in <math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k.</math>
*[Waring's problem](/source/Waring's_problem) asks whether for every natural number {{mvar|k}} there exists an associated positive integer {{mvar|s}} such that every natural number is the sum of at most {{mvar|sk}}<sup>th</sup> powers of natural numbers.
*The successive powers of the [golden ratio](/source/Golden_ratio) ''φ'' obey the Fibonacci recurrence: <math display=block>
\varphi^{n+1} = \varphi^n + \varphi^{n-1}.</math>
*[Newton's identities](/source/Newton's_identities) express the sum of the {{mvar|k}}<sup>th</sup> 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](/source/Cube_(algebra)) is sometimes another cube.
*The [Fermat cubic](/source/Fermat_cubic), in which the sum of three cubes equals another cube, has a general solution.
*The [power sum symmetric polynomial](/source/power_sum_symmetric_polynomial) is a building block for symmetric polynomials.
*The [sum of the reciprocals of all perfect powers](/source/Perfect_power) including duplicates (but not including 1) equals 1.
*The [Erdős–Moser equation](/source/Erd%C5%91s%E2%80%93Moser_equation), <math>1^k+2^k+\cdots+m^k=(m+1)^k</math> where {{mvar|m}} and {{mvar|k}} are positive integers, is conjectured to have no solutions other than {{nowrap|1=1<sup>1</sup> + 2<sup>1</sup> = 3<sup>1</sup>}}.
*The [sums of three cubes](/source/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](/source/geometric_series) is <math>\sum_{i=k}^{n} z^i = \frac{z^{k}-z^{n+1}}{1-z}.</math>
*The sum of powers can be expressed as such : <math>\sum_{i=1}^{n} i^k = \sum_{i=1}^{n}\sum_{j=i}^{n} j^{k-a}(i^a-(i-1)^a)</math> for n greater than 1, a greater than 1 and all k. This one is useful to help determine subsequent sums of power based on known previous sums of power.

==See also==

* [Sum of squares](/source/Sum_of_squares)
* [Sum of reciprocals](/source/Sum_of_reciprocals)
* [Diophantine equation](/source/Diophantine_equation)

== References ==
{{reflist|30em}}

Category:Number theory
Category:Mathematics-related lists
Category:Squares in number theory

{{sia}}

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