Sums of powers

In mathematics and statistics, sums of powers occur in a number of contexts:

• Sums of squares arise in many contexts.
• Faulhaber's formula expresses $1^k + 2^k + 3^k + \cdots + n^k$ as a polynomial in n.
• Fermat's right triangle theorem states that there is no solution in positive integers for $a^4=b^4+c^2.$
• Fermat's Last Theorem states that $x^k+y^k=z^k$ is impossible in positive integers with k>2.
• The equation of a superellipse is $|x/a|^k+|y/b|^k=1$. 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 $a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4$ 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 $\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k.$
• 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 s k^th powers of natural numbers.
• The successive powers of the golden ratio φ obey the Fibonacci recurrence:

$\varphi^{n+1} = \varphi^n + \varphi^{n-1}.$

• 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.