positive integer

Mathematical terminology

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Like integers, positive integers are also Countable Of Infinite set 。 stay number theory Middle, positive integer, i.e. 1, 2, 3; But in set theory and computer science The natural number usually refers to Nonnegative integer , that is, positive integer and 0 aggregate It can also be said that natural numbers other than 0 are positive integers. Positive integers can also be divided into prime numbers, 1 and composite numbers. Positive integers can have a positive sign (+) or not.

Chinese name: positive integer
Foreign name: Positive integer
Category: integer

Comparison: negtive integer
Classification: 1. Prime number, composite number
Theorem: Basic theorem of arithmetic Discrete inequality
Definition: Integer greater than 0

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▪ Discrete inequality
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definition

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Positive integer, which is an integer greater than 0, also Positive number With integers intersection 。 Positive integers can also be divided into prime numbers, 1 and composite numbers. Positive integers can have a positive sign (+) or not. For example,+1,+6, 3, and 5 are positive integers. 0 is neither a positive integer nor negtive integer (0 is an integer).^[1]

The earliest number in human history is natural number (positive integer).^[5]

Integer classification

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Integers are divided into three categories with 0 as the boundary:

1. Positive integer, that is, integer greater than 0, such as 1, 2, 3

2. 0 is neither a positive integer nor negtive integer (0 is an integer).

3. Negative integer, that is, integer less than 0, such as - 1, - 2, - 3^[1]

Positive integer classification

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One way to classify positive integers is by their Divisor Or the number of product factors. For example, if there are only two, it is called Prime number Or prime numbers, and more than two are called Composite number 。^[2]

Peano axioms

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utilize Peano axioms Integer and N *It is described as follows:

Any non empty set that meets the following conditions is called a positive integer aggregate , recorded as N *。 If

I1 is a positive integer;

Ⅱ Each positive integer determined a , have a certain number of successors a ' ， a 'is also a positive integer (number a Successors of a 'is the integer immediately following this number（ a +1）。 For example, 1 '=2, 2'=3, etc.);

Ⅲ If b 、 c All positive integers a The number of successors of, then b = c ；

Ⅳ 1 is not the successor of any positive integer;

Ⅴ Design S ⊆ N *, and satisfies two conditions (i) 1 ∈ S ； (ii) If n ∈ S , then n '∈ S 。 that S Is a set of all positive integers, that is S = N *。 (This axiom is also called Inductive axiom , guaranteed Mathematical induction Correctness of)

Peano axioms yes N *Has been characterized and appointment , from which we can deduce various kinds of positive integers nature 。^[3]

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Basic theorem of arithmetic

The unique decomposition theorem of positive integers: also called Basic theorem of arithmetic 。

That is, every natural number greater than 1 can be written as the product of the powers of several prime numbers, and the writing method is unique after these prime factors are arranged in size.

Discrete inequality

if X ， N ∈ N *, then X > N Equivalent to X ≥ N +1。

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On the Hexagonal Number Part of Positive Integers

For any positive number n, let b (n) represent the maximum hexagon number part of n, that is, b (n)=m (2m-1). If m (2m-1) ≤ n<(m+1) (2m+1), m ∈ N.^[4]

The Combined Representation of the k-th Power of the First n Positive Integers

Use several shapes such as

Find the expansion form of

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