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, or alternatively in term of a Bernoulli polynomial.
• 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 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 ${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$ in integers.
• The Prouhet–Tarry–Escott problem considers sums of two sets of 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 ${\displaystyle \sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}.}$
• 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 k^th powers of natural numbers.
• The successive powers of the golden ratio φ obey the Fibonacci recurrence:

${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}$

• 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.
• The Erdős–Moser equation, ${\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}}$ where ${\displaystyle m}$ and ${\displaystyle k}$ 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 S_m(z, n) = z^m + (z+1)^m + ... + (z+n−1)^m is related to the Bernoulli polynomials B_m(z) by (∂_n−∂_z) S_m(z, n) = B_m(z) and (∂_2λ−∂_Z) S_2k+1(z, n) = Ŝ′_k+1(Z) where Z = z(z−1), λ = S₁(z, n), Ŝ_k+1(Z) ≡ S_2k+1(0, z).[1]
• the sum of the terms in the geometric series is ${\displaystyle \sum _{k=i}^{n}z^{k}={\frac {z^{i}-z^{n+1}}{1-z}}}$

See also

• Diophantine equation

References

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

• Reznick, Bruce and Rouse, J. "On the sums of two cubes", Int. J. Number Theory, 7 (2011), 1863–1882, MR2854220.