Inmathematics andstatistics,sums of powers occur in a number of contexts:

• Sums of squares arise in many contexts. For example, ingeometry, thePythagorean theorem involves the sum of two squares; innumber theory, there areLegendre's three-square theorem andJacobi's four-square theorem; and instatistics, theanalysis of variance involves summing the squares of quantities.
• There are only finitely many positive integers that are not sums ofdistinct 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${\displaystyle 1^k+2^k+3^k+\cdots +n^k}$ as a polynomial inn, or alternativelyin terms of a Bernoulli polynomial.
• Fermat's right triangle theorem states that there is no solution in positive integers for${\displaystyle a^2=b^4+c^4}$ and${\displaystyle a^4=b^4+c^2}$.
• Fermat's Last Theorem states that${\displaystyle x^k+y^k=z^k}$ is impossible in positive integers withk > 2.
• The equation of asuperellipse is${\displaystyle |x/a{|}^k+|y/b{|}^k=1}$. Thesquircle is the casek = 4,a =b.
• Euler's sum of powers conjecture (disproved) concerns situations in which the sum ofn integers, each ak^th power of an integer, equals anotherk^th power.
• TheFermat-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.
• TheJacobi–Madden equation is${\displaystyle a^4+b^4+c^4+d^4=(a+b+c+d{)}^4}$ in integers.
• TheProuhet–Tarry–Escott problem considers sums of two sets ofk^th powers of integers that are equal for multiple values ofk.
• Ataxicab number is the smallest integer that can be expressed as a sum of two positive third powers inn distinct ways.
• TheRiemann zeta function is thesum of reciprocals of the positive integers each raised to the powers, wheres is acomplex number whose real part is greater than 1.
• TheLander, Parkin, and Selfridge conjecture concerns the minimal value ofm +n in${\displaystyle \sum _{i=1}^n{a}_i^k=\sum _{j=1}^m{b}_j^k.}$
• Waring's problem asks whether for every natural numberk there exists an associated positive integers such that every natural number is the sum of at mostsk^th powers of natural numbers.
• The successive powers of thegolden ratioφ obey the Fibonacci recurrence:$${\displaystyle {\phi }^{n+1}={\phi }^n+{\phi }^{n-1}.}$$
• Newton's identities express the sum of thek^th powers of all the roots of a polynomial in terms of the coefficients in the polynomial.
• Thesum of cubes of numbers in arithmetic progression is sometimes another cube.
• TheFermat cubic, in which the sum of three cubes equals another cube, has a general solution.
• Thepower sum symmetric polynomial is a building block for symmetric polynomials.
• Thesum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1.
• TheErdős–Moser equation,${\displaystyle 1^k+2^k+\cdots +m^k=(m+1{)}^k}$ wherem andk are positive integers, is conjectured to have no solutions other than1¹ + 2¹ = 3¹.
• Thesums 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 thegeometric series is${\displaystyle \sum _{i=k}^n{z}^i=\frac{z^k-z^{n+1}}{1-z}.}$

• Sum of squares
• Sum of reciprocals
• Diophantine equation

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