Generalized taxicab number

Last updated
Unsolved problem in mathematics:

Does there exist any number that can be expressed as a sum of two positive fifth powers in at least two different ways, i.e., ?

Contents

In number theory, the generalized taxicab numberTaxicab(k, j, n) is the smallest number — if it exists — that can be expressed as the sum of j numbers to the kth positive power in n different ways. For k = 3 and j = 2, they coincide with taxicab number.

The latter example is 1729, as first noted by Ramanujan.

Euler showed that

However, Taxicab(5, 2, n) is not known for any n 2:
No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists. [1]

The largest variable of must be at least 3450.[ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Antiderivative</span> Concept in calculus

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.

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 kth powers of positive integers is itself a kth power, then n is greater than or equal to k:

<span class="mw-page-title-main">Square-free integer</span> Number without repeated prime factors

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e.

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

<span class="mw-page-title-main">Lagrange's four-square theorem</span> Every natural number can be represented as the sum of four integer squares

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as a sum of four non-negative integer squares. That is, the squares form an additive basis of order four.

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

1729 is the natural number following 1728 and preceding 1730. It is notably the first nontrivial taxicab number.

<span class="mw-page-title-main">Taxicab number</span> Smallest integer expressable as a sum of two positive integer cubes in n distinct ways

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy-Ramanujan number.

In mathematics, a harshad number in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

In number theory, the n-th cabtaxi number, typically denoted Cabtaxi(n), is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways. Such numbers exist for all n, which follows from the analogous result for taxicab numbers.

In number theory, a self number or Devlali number in a given number base is a natural number that cannot be written as the sum of any other natural number and the individual digits of . 20 is a self number, because no such combination can be found. 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.

The Beal conjecture is the following conjecture in number theory:

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence of non-negative real numbers, the series converges if and only if the "condensed" series converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.

In mathematical logic, Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely.

In mathematics and statistics, sums of powers occur in a number of contexts:

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 additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order. More specifically, it states that for every fixed integer , there exists a subset of the natural numbers satisfying

<span class="mw-page-title-main">Sum of two cubes</span> Mathematical polynomial formula

In mathematics, the sum of two cubes is a cubed number added to another cubed number.

References

  1. Guy, Richard K. (2004). Unsolved Problems in Number Theory (Third ed.). New York, New York, USA: Springer-Science+Business Media, Inc. ISBN   0-387-20860-7.