Sums of powers

Last updated December 12, 2024

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.
There are only finitely many positive integers that are not sums of distinct squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See ^[1] for a generalization to sums of polynomials.
Faulhaber's formula expresses $1^{k}+2^{k}+3^{k}+\cdots +n^{k}$ as a polynomial in $n$ , or alternatively in terms 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 $\sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}.$
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 $sk$ ^th powers of natural numbers.
The successive powers of the golden ratio φ obey the Fibonacci recurrence: $\varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.$
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, $1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}$ 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 sum of the terms in the geometric series is $\sum _{i=k}^{n}z^{i}={\frac {z^{k}-z^{n+1}}{1-z}}.$

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In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted $i$ , called the imaginary unit and satisfying the equation $; every complex number can be expressed in the form, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number, a is called the real part, and b is called the imaginary part . The set of complex numbers is denoted by either of the symbols or C . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.$

<span class="mw-page-title-main">Diophantine equation</span> Polynomial equation whose integer solutions are sought

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers $n$ and $k$ greater than 1, if the sum of $n$ many $k$ th powers of positive integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ :

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod $p$ when $p$ is a prime number.

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

<span class="mw-page-title-main">Euler's totient function</span> Number of integers coprime to and less than n

In number theory, Euler's totient function counts the positive integers up to a given integer $n$ that are relatively prime to $n$ . It is written using the Greek letter phi as $or, and may also be called Euler's phi function . In other words, it is the number of integers k in the range 1 \leq k \leq n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n .$

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

In mathematics, a Fermat number, named after Pierre de Fermat (1607–1665), the first known to have studied them, is a positive integer of the form: $where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ....$

<span class="mw-page-title-main">Root of unity</span> Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power $n$ . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In modular arithmetic, a number $g$ is a primitive root modulo $n$ if every number $a$ coprime to $n$ is congruent to a power of $g$ modulo $n$ . That is, $g$ is a primitive root modulo $n$ if for every integer $a$ coprime to $n$ , there is some integer $k$ for which $g$ ^$k$ ≡ $a$ . Such a value $k$ is called the index or discrete logarithm of $a$ to the base $g$ modulo $n$ . So $g$ is a primitive root modulo $n$ if and only if $g$ is a generator of the multiplicative group of integers modulo $n$ .

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:

The Beal conjecture is the following conjecture in number theory:

In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function $is called refinable with respect to the mask if$

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to $, the field of rational numbers.$

In number theory, the Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some $k$ -th powers equals the sum of some other $k$ -th powers, then the total number of terms in both sums combined must be at least $k$ .

In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers $a$ and $b$ to equal the reciprocal of a third positive integer $c$ :

This is a glossary of concepts and results in number theory, a field of mathematics. Concepts and results in arithmetic geometry and diophantine geometry can be found in Glossary of arithmetic and diophantine geometry.

References

↑ Graham, R. L. (June 1964). "Complete sequences of polynomial values". Duke Mathematical Journal. 31 (2): 275–285. doi:10.1215/S0012-7094-64-03126-6. ISSN 0012-7094.

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[1] Graham, R. L. (June 1964). "Complete sequences of polynomial values". Duke Mathematical Journal. 31 (2): 275–285. doi:10.1215/S0012-7094-64-03126-6. ISSN 0012-7094.

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Sums of powers

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