In mathematics, the sum of two cubes is a cubed number added to another cubed number.
Every sum of cubes may be factored according to the identity
Binomial numbers are the general of this factorization to higher odd powers.
The mnemonic "SOAP", standing for "Same, Opposite, Always Positive", is sometimes used to memorize the correct placement of the addition and subtraction symbols while factorizing cubes. [2] When applying this method to the factorization, "Same" represents the first term with the same sign as the original expression, "Opposite" represents the second term with the opposite sign as the original expression, and "Always Positive" represents the third term and is always positive.
original sign | Same | Opposite | Always Positive | |||||
Starting with the expression, is multiplied by a and b [1]
By distributing a and b to , [1]
and by canceling the alike terms, [1]
Similarly for the difference of cubes,
Fermat's last theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer cube. The first recorded proof of the exponent 3 case was given by Euler. [3]
Taxicab numbers are numbers that can be expressed as a sum of two positive integer cubes in n distinct ways. The smallest taxicab number, after Ta(1), is 1729, [4] expressed as
The smallest taxicab number expressed in 3 different ways is 87,539,319, expressed as
Cabtaxi numbers are numbers that can be expressed as a sum of two positive or negative integers or 0 cubes in n ways. The smallest cabtaxi number, after Cabtaxi(1), is 91, [5] expressed as:
The smallest Cabtaxi number expressed in 3 different ways is 4104, [6] expressed as
In number theory, a Carmichael number is a composite number , which in modular arithmetic satisfies the congruence relation:
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:
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4.
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
15 (fifteen) is the natural number following 14 and preceding 16.
1729 is the natural number following 1728 and preceding 1730. It is notably the first nontrivial taxicab number.
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.
126 is the natural number following 125 and preceding 127.
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence
216 is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato.
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 mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
In mathematics and statistics, sums of powers occur in a number of contexts:
1105 is the natural number following 1104 and preceding 1106.