Sums of powers

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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 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.
  • The Erdős–Moser equation, where and are positive integers, is conjectured to have no solutions other than 11 + 21 = 31.
  • The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown whether all remaining integers can be expressed in this form.
  • The sums of powers Sm(z,n) = zm+(z+1)m+...+(z+n-1)m is related to the Bernoulli polynomials Bm(z) by (∂n-∂z) Sm(z,n) = Bm(z) and (∂-∂Z) S2k+1(z,n) = Ŝ'k+1(Z) where Z = z(z-1), λ = S1(z,n), Ŝk+1(Z) ≡ S2k+1(0,z). [1]

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References[edit]

  1. ^ Do Tan Si. "The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked". Applied Mathematics 10.03 (2019): 100-112. Scientific Research.