Positive integer in the context of Canonical form

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

A Pythagorean triple consists of three positive integers a, b, and c, such that a + b = c. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle.

A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing (a, b, c) by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements).

In mathematics, a ratio (/ˈreɪ.ʃ(i.)oʊ/) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.

A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself.

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".

In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers ${\displaystyle 1,2,...,n^{2}}$, the magic square is said to be normal. Many authors take magic square to mean normal magic square.

Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square).

An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.

Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of positive integers), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.

In mathematics, an nth root of a number x is a number r which, when raised to the power of n, yields x: $${\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.}$$ The positive integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an nth root is a root extraction.

The nth root of x is written as ${\displaystyle {\sqrt[{n}]{x}}}$ using the radical symbol ${\displaystyle {\sqrt {\phantom {x}}}}$. The square root is usually written as ⁠${\displaystyle {\sqrt {x}}}$⁠, with the degree omitted. Taking the nth root of a number, for fixed ⁠${\displaystyle n}$⁠, is the inverse of raising a number to the nth power, and can be written as a fractional exponent:

In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4). Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem.

To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involves testing whether each factor is prime each time a factor is found.

A milestone is a numbered marker placed on a route such as a road, railway line, canal or boundary. They can indicate the distance to towns, cities, and other places or landmarks like mileage signs; or they can give their position on the route relative to some datum location (a zero milepost). On roads they are typically located at the side or in a median or central reservation. They are alternatively known as mile markers (sometimes abbreviated MMs), mileposts or mile posts (sometimes abbreviated MPs). A "kilometric point" is a term used in metricated areas, where distances are commonly measured in kilometres instead of miles. "Distance marker" is a generic unit-agnostic term.

Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between ${\displaystyle n^{2}}$ and ${\displaystyle (n+1)^{2}}$ for every positive integer ${\displaystyle n}$. The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠${\displaystyle \textstyle (x+y)^{n}}$⁠ expands into a polynomial with terms of the form ⁠${\displaystyle \textstyle ax^{k}y^{m}}$⁠, where the exponents ⁠${\displaystyle k}$⁠ and ⁠${\displaystyle m}$⁠ are nonnegative integers satisfying ⁠${\displaystyle k+m=n}$⁠ and the coefficient ⁠${\displaystyle a}$⁠ of each term is a specific positive integer depending on ⁠${\displaystyle n}$⁠ and ⁠${\displaystyle k}$⁠. For example, for ⁠${\displaystyle n=4}$⁠,$${\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}$$

The coefficient ⁠${\displaystyle a}$⁠ in each term ⁠${\displaystyle \textstyle ax^{k}y^{m}}$⁠ is known as the binomial coefficient ⁠${\displaystyle {\tbinom {n}{k}}}$⁠ or ⁠${\displaystyle {\tbinom {n}{m}}}$⁠ (the two have the same value). These coefficients for varying ⁠${\displaystyle n}$⁠ and ⁠${\displaystyle k}$⁠ can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where ⁠${\displaystyle {\tbinom {n}{k}}}$⁠ gives the number of different combinations (i.e. subsets) of ⁠${\displaystyle k}$⁠ elements that can be chosen from an ⁠${\displaystyle n}$⁠-element set. Therefore ⁠${\displaystyle {\tbinom {n}{k}}}$⁠ is usually pronounced as "⁠${\displaystyle n}$⁠ choose ⁠${\displaystyle k}$⁠".