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.

Sums of powers

See also