Sums of powers

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.
• Faulhaber's formula expresses ${\displaystyle 1^{k}+2^{k}+3^{k}+\cdots +n^{k}}$ 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 ${\displaystyle a^{2}=b^{4}+c^{4}}$ 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 with k > 2.
• The equation of a superellipse is ${\displaystyle |x/a|^{k}+|y/b|^{k}=1}$. 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 k^th power of an integer, equals another k^th 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 ${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$ in integers.
• The Prouhet–Tarry–Escott problem considers sums of two sets of k^th 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 ${\displaystyle \sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}.}$
• 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 sk^th powers of natural numbers.
• The successive powers of the golden ratio φ obey the Fibonacci recurrence:
  $${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}$$

  
• Newton's identities express the sum of the k^th 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, ${\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}}$ 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 sums of powers
  $${\displaystyle S_{m}(z,n)=z^{m}+(z+1)^{m}+\dots +(z+n-1)^{m}}$$

  
  is related to the Bernoulli polynomials B_m(z) by
  $${\displaystyle {\begin{aligned}(\partial _{n}-\partial _{z})S_{m}(z,n)&=B_{m}(z),\\[4pt](\partial _{2\lambda }-\partial _{Z})S_{2k+1}(z,n)&={\hat {S}}'_{k+1}(Z),\end{aligned}}}$$

  
  where ${\displaystyle Z=z(z-1),}$ ${\displaystyle \lambda =S_{1}(z,n),}$ ${\displaystyle {\hat {S}}_{k+1}(Z)\equiv S_{2k+1}(0,z).}$
• The sum of the terms in the geometric series is ${\displaystyle \sum _{k=i}^{n}z^{k}={\frac {z^{i}-z^{n+1}}{1-z}}.}$