0_Wizard
Hi there! This is a funny question but what is your favorite number? Mine is 7 which is the natural number following 6 and right before 8
7 (seven; ) is the natural number following 6 and preceding 8.
chosen_7_0
7
chosen_topic_0_7
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying 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, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. 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.
Wieferich prime
In number theory, a Wieferich prime is a prime number "p" such that "p" divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime "p" divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime.
Natural number
In mathematics, the natural numbers are those used for counting (as in "there are "six" coins on the table") and ordering (as in "this is the "third" largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with 1, corresponding to the positive integers . Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.
Mathematical induction
Mathematical induction is a mathematical proof technique, a form of direct proof, usually done in two steps. It is used to prove a given statement about any well-ordered set. Most commonly, the well-ordered set is the set of natural numbers. The first step, called the base case or basis, proves that the theorem is true for the number one. The second step, called the inductive step, proves that, if the theorem is true for a given number, then it is also true for the next number. These two steps establish the theorem for all natural numbers. The fact that these two steps prove that a theorem is true in an infinite number of cases is called the "Principle of mathematical induction".
0
0 (zero; ) is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought (UK), naught (US) (), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (). Informal or slang terms for zero include zilch and zip. "Ought" and "aught" (), as well as "cipher", have also been used historically.
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers.
1_Apprentice
Mine happens to be 7 as well! Many seems to associate 7 as being a lucky number.
0_Wizard
It is a prime number so it cannot be formed by multiplying two smaller natural numbers
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
self_Prime_number_0
Prime number
self_0_Prime_number
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying 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, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. 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.
Euclidean algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's "Elements" (c. 300 BC). It is an example of an "algorithm", a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Proof by infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle. One typical application is to show that a given equation has no solutions. Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction (repeating the same step) the original premise—that any solution exists— is incorrect: its correctness produces a contradiction.
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4 and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, "number" may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as and , real numbers such as and , and complex numbers, which extend the real numbers by adding a square root of.
Mathematical induction
Mathematical induction is a mathematical proof technique, a form of direct proof, usually done in two steps. It is used to prove a given statement about any well-ordered set. Most commonly, the well-ordered set is the set of natural numbers. The first step, called the base case or basis, proves that the theorem is true for the number one. The second step, called the inductive step, proves that, if the theorem is true for a given number, then it is also true for the next number. These two steps establish the theorem for all natural numbers. The fact that these two steps prove that a theorem is true in an infinite number of cases is called the "Principle of mathematical induction".
Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity. Interpreted as nimbers, ordinals are also subject to nimber arithmetic operations. The union of two disjoint well-ordered sets "S" and "T" can be well-ordered.
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.
1_Apprentice
Yes, there are various other prime numbers as well!
0_Wizard
Right! There are an infinite number of prime numbers.
By Euclid's theorem, there are an infinite number of prime numbers.
partner_List_of_prime_numbers_1
List of prime numbers
partner_5_List_of_prime_numbers
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying 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, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. 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.
Infinity
Infinity (symbol: ) is a concept describing something "without any bound" or "larger than any natural number". Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea also is used in physics and the other sciences. In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4 and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, "number" may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as and , real numbers such as and , and complex numbers, which extend the real numbers by adding a square root of.
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel, or simply Hilbert's Hotel, is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche" reprinted in and was popularized through George Gamow's 1947 book "One Two Three... Infinity". Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied.
Euclidean algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's "Elements" (c. 300 BC). It is an example of an "algorithm", a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem. Euclid offered a proof published in his work "Elements" (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers "p", "p", ..., "p". It will be shown that at least one additional prime number not in this list exists. Let "P" be the product of all the prime numbers in the list: "P" = "p""p"..."p". Let "q" = "P" + 1. Then "q" is either prime or not:   This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers.
Pi
The number () is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "" since the mid-18th century, though it is also sometimes spelled out as "pi". Being an irrational number, cannot be expressed exactly as a common fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate .
1_Apprentice
I did not know that, I just know of a few prime numbers, like 2, 3, 5, 7, 11, and 13
0_Wizard
According to Euclid, there are infinitely many prime numbers! 
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.
self_Euclid's_theorem_0
Euclid's theorem
self_5_Euclids_theorem
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying 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, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. 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.
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem. Euclid offered a proof published in his work "Elements" (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers "p", "p", ..., "p". It will be shown that at least one additional prime number not in this list exists. Let "P" be the product of all the prime numbers in the list: "P" = "p""p"..."p". Let "q" = "P" + 1. Then "q" is either prime or not:   This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers.
Prime quadruplet
A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {"p", "p"+2, "p"+6, "p"+8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. The first eight prime quadruplets are:  All prime quadruplets except {5, 7, 11, 13} are of the form {30"n" + 11, 30"n" + 13, 30"n" + 17, 30"n" + 19} for some integer "n". (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade. A prime quadruplet contains two pairs of twin primes or can be described as having two overlapping prime triplets.
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet "L"-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique.
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers "a" and "d", there are infinitely many primes of the form "a" + "nd", where n is a non-negative integer. In other words, there are infinitely many primes that are congruent to "a" modulo "d". The numbers of the form "a" + "nd" form an arithmetic progression  and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes.
Euclid number
In mathematics, Euclid numbers are integers of the form , where "p"# is the "n"th primorial, i.e. the product of the first "n" prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers. For example, the first three primes are 2, 3, 5; their product is 30, and the corresponding Euclid number is 31. The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... . It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers.
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . If is a composite number then so is . ( is divisible by both and .) This definition is therefore equivalent to a definition as a prime number of the form for some prime . More generally, numbers of the form without the primality requirement are called Mersenne numbers.