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 as a polynomial in n.
- Fermat's right triangle theorem states that there is no solution in positive integers for
- Fermat's Last Theorem states that is impossible in positive integers with k>2.
- The equation of a superellipse is . The squircle is the case .
- 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 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
- 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:
- Newton's identities express the sum of the kth 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.
See also
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