Cross slope

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 Last Theorem states that $x^k+y^k=z^k$ is impossible in positive integers with k>2.
• 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:

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