# 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.
• Fermat's right triangle theorem states that there is no solution in positive integers for ${\displaystyle a^{4}=b^{4}+c^{2}.}$
• 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 ${\displaystyle k=4,a=b}$.
• Euler's sum of powers conjecture (disproved) concerns situations in which the sum of n integers, each a kth power of an integer, equals another kth 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 kth 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 s kth powers of natural numbers.
• The successive powers of the golden ratio φ obey the Fibonacci recurrence:
${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}$