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.
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
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.
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 skth powers of natural numbers.
The successive powers of the
golden ratioφ 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.
This
article includes a list of related items that share the same name (or similar names). If an
internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.
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.
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
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.
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 skth powers of natural numbers.
The successive powers of the
golden ratioφ 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.
This
article includes a list of related items that share the same name (or similar names). If an
internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.