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 $1^k + 2^k + 3^k + \cdots + n^k$ 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 $a^2=b^4+c^4$ and $a^4=b^4+c^2$.
*Fermat's Last Theorem states that $x^k+y^k=z^k$ is impossible in positive integers with ''k''>2.
*The equation of a superellipse is $, x/a, ^k+, y/b, ^k=1$. The squircle is the case $k=4, a=b$.
*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 $a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4$ 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 $\sum_^ a_i^k = \sum_^ b_j^k.$
*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:

::$\varphi^ = \varphi^n + \varphi^.$

*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, $1^k+2^k+\cdots+m^k=(m+1)^k$ where $m$ and $k$ 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''_2''k''+1(''z'', ''n'') = ''Ŝ''′_''k''+1(''Z'') where ''Z'' = ''z''(''z''−1), λ = ''S''₁(''z'', ''n''), ''Ŝ''_''k''+1(''Z'') ≡ ''S''_2''k''+1(0, ''z'').
*The sum of the terms in the geometric series is $\sum_^ z^k = \frac.$

See also

* Sum of squares
* Sum of reciprocals
* Diophantine equation

Number theory
Mathematics-related lists
Squares in number theory

References

* {{cite journal , arxiv=1012.5801 , mr=2854220, doi=10.1142/S1793042111004903 , title=On the Sums of Two Cubes , year=2011 , last1=Reznick , first1=Bruce , author1-link=Bruce Reznick , last2=Rouse , first2=Jeremy , journal=International Journal of Number Theory , volume=07 , issue=7 , pages=1863–1882 , s2cid=16334026