hcju/proofwikips huggingface dataset:
Definition:Central Field
Let $R$ be a simply connected region in $\paren {n + 1}$-dimensional space. Let $\tuple {x, \mathbf y}$ be a point in $R$. Let $c = \tuple {\sequence {c_i}_{0 \mathop \le i \mathop \le n} }$ be a point lying outside of $R$. Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ whose extremals $\mathbf y$ are curves in $\paren{n+1}$-dimensional space. {{explain|Extremum or extremal, or are they the same thing? If they are, then an "also known as" section needs to be placed on the page Definition:Extremum of Functional. If not, then the terminology needs to be specifically amended and/or corrected.}} Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\tuple {x, \mathbf y}$, thereby for every point in $R$ defining a relation: :$(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$. Then the field of directions $(1)$ is called a '''central field'''.
Definition:Time/Unit/Quarter-Year
The '''quarter-year''' is defined as: :$91$ days or :$13$ weeks of $7$ days each.
Definition:Fermat Pseudoprime/Base 3
Let $q$ be a composite number such that $3^q \equiv 3 \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $3$'''.
Definition:Celsius
'''Celsius''' is a temperature scale. Its two reference points are: : $0 \cels$, which is set at the melting point of water. : $100 \cels$, which is set at the boiling point of water, as defined at sea level and standard atmospheric pressure. A temperature measured in '''Celsius''' is often referred to as so many '''degrees Celsius'''. === Symbol === {{:Definition:Celsius/Symbol}}
Definition:Celsius/Symbol
The symbol for the '''degree Celsius''' is $\cels$.
Definition:Array/Row
Let $\mathbf A$ be an array. The '''rows''' of $\mathbf A$ are the lines of elements reading '''across''' the page.
Definition:Array/Element/Index
Let $\mathbf A$ be an $m \times n$ array. Let $a_{i j}$ be an element of $\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$.
Definition:Array/Element
Let $\mathbf A$ be an Array. The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the '''elements''' of $\mathbf L$. The '''element''' at row $i$ and column $j$ is called '''element $\tuple {i, j}$ of $\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \left({i, j}\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the array where the '''element''' is positioned.
Definition:Array/Square
An array whose dimensions are equal is called a '''square array'''. That is, a '''square array''' is an array which has the same number of rows as it has columns. A '''square array''' $\left[{a}\right]_{n n}$ is usually denoted $\left[{a}\right]_n$.
Definition:Prime Magic Square
A '''prime magic square''' is a magic square whose elements are all prime numbers, and may or may not also include the number $1$.
Definition:Extremal Embedding in Field of Functional
Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let $\gamma$ be an extremal of $J$. {{explain|Extremum or extremal?}} Let $R$ be a simply connected open region which contains $\gamma$ as a subset. {{explain|GF puts "open" in brackets; does it matter?}} Let a field of functional $J$ be defined at every point of $R$. Let one of the trajectories of the field be $\gamma$. Then $\gamma$ can be '''embedded''' in a field of functional $J$. {{explain|If $\gamma$ "can be" embedded, is it correct to say that $\gamma$ "is" embedded? If not, then is it more grammatically accurate to define $\gamma$ as being "embeddable"? Otherwise it looks as though "can be" introduces a statement that needs to be proven.}}
Definition:Permutable Prime
A '''permutable prime''' is a prime number $p$ which has the property that all anagrams of $p$ are prime.
Definition:Permutable Prime/Sequence
The sequence of '''permutable primes''' begins: :$2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R_{19}, R_{23}, R_{317}, R_{1091}, \ldots$ where $R_n$ denotes the repunit of $n$ digits.
Definition:Anagram
Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $m$ be an integer expressed in base $b$. An '''anagram base $b$''' of $m$ is an integer formed by the digits of $m$ written in a different order. When the number base of $m$ is not specified, base $10$ is assumed.
Definition:Repunit Prime
Let $b \in \Z_{>1}$ be an integer greater than $1$. A '''repunit prime base $b$''' is a repunit base $b$ which is prime.
Definition:Repunit Prime/Index
The '''index''' of a '''repunit prime''' is the number of digits it has. Thus $R_n$ denotes a '''repunit prime''' with $n$ digits.
Definition:Repunit/Index
The '''index''' of a '''repunit''' is the number of digits it has. Thus a '''repunit''' with $n$ digits can be referred to as $R_n$.
Definition:Fermat Set
The '''Fermat set''' is the set of integers: :$\left\{ {1, 3, 8, 120}\right\}$ It has the property of being a Diophantine quadruple.
Definition:Diophantine m-Tuple/Quadruple
A '''Diophantine quadruple''' is a Diophantine $m$-tuple where $m = 4$. That is, a '''Diophantine quadruple''' is a set of $4$ positive integers such that the product of any $2$ of them plus $1$ is a square number.
Definition:Diophantine m-Tuple
A '''Diophantine $m$-tuple''' set of $m$ positive integers such that the product of any $2$ of them plus $1$ is a square number.
Definition:Diophantine m-Tuple/Quintuple
A '''Diophantine quintuple''' is a Diophantine $m$-tuple where $m = 5$. That is, a '''Diophantine quintuple''' is a set of $5$ positive integers such that the product of any $2$ of them plus $1$ is a square number.
Definition:Triperfect Number
A '''triperfect number''' is a positive integer $n$ such that the sum of its divisors is equal to $3$ times $n$.
Definition:Multiply Perfect Number
A '''multiply perfect number''' is a positive integer $n$ such that the sum of its divisors is equal to an integer multiple of $n$.
Definition:Hilbert's Invariant Integral
Let $\mathbf y$ be an $n$-dimensional vector. Let $H$ be Hamiltonian and $\mathbf p$ momenta. Let $\Gamma$ be a curve connecting points $\tuple {x_0, \map{\mathbf y} {x_0} }$ and $\tuple {x, \mathbf y}$. Then the following line integral is known as '''Hilbert's Invariant Integral''': :$\displaystyle \map g {x, \mathbf y} = \int_\Gamma \paren {-H \rd x + \mathbf p \rd \mathbf y}$ {{NamedforDef|David Hilbert|cat = Hilbert}}
Definition:Weierstrass E-Function
Let $\mathbf y, \mathbf z, \mathbf w : \R \to \R^n$ be $n$-dimensional vector-valued functions. Let $\mathbf y$ be such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$. Let $F: \R^{2 n + 1} \to \R$ be twice differentiable {{WRT|Differentiation}} its (independent) variables. Let $J$ be a functional such that: :$\ds J \sqbrk{\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where:  :$\mathbf y' := \dfrac {\d \mathbf y} {\d x}$ is the derivative of a vector-valued function. === Definition 1 === {{:Definition:Weierstrass E-Function/Definition 1}} === Definition 2 === {{:Definition:Weierstrass E-Function/Definition 2}}
Definition:Multiply Perfect Number/Order
Let $n \in \Z_{>0}$ be a '''multiply perfect number''' such that the sum of its divisors is equal to $m \times n$. Then $n$ is '''multiply perfect of order $m$'''.
Definition:Wonderful Demlo Number
The '''Wonderful Demlo numbers''' are the positive integers defined as the coefficients of the powers of $x$ in the expansion of: :$\dfrac {1 + 10 x} {\left({1 - x}\right) \left({1 - 10 x}\right) \left({1 - 100 x}\right)}$
Definition:Wonderful Demlo Number/Sequence
The sequence of '''Wonderful Demlo numbers''' begins: :$1, 121, 12 \, 321, 1 \, 234 \, 321, 123 \, 454 \, 321, \ldots$
Definition:Obstinate Number
An '''obstinate number''' is a positive odd integer which cannot be expressed as the sum of a power of $2$ and a prime.
Definition:Ore Number
Let $n \in \Z_{>0}$ be a positive integer. $n$ is an '''Ore number'''  the harmonic mean of its divisors is an integer.
Definition:Woodall Number
A '''Woodall number''' is a positive integer of the form: :$n \times 2^n - 1$
Definition:Primitive Prime Factor of Fibonacci Number
Let $F_n$ denote the $n$th Fibonacci number. A '''primitive prime factor of $F_n$''' is a prime number $p$ of $F_n$ such that: :$p \divides F_n$ :$\nexists k \in \Z_{>0}: k < n: p \divides F_k$ where $a \divides b$ denotes that $a$ is a divisor of $b$. That is, a prime factor of $F_n$ but of no smaller Fibonacci numbers.
Definition:Primitive Prime Factor
Let $\left\langle{a_n}\right\rangle$ be an integer sequence. A '''primitive prime factor''' for a term $a_n$ is a prime number $p$ of $a_n$ such that: :$p \divides a_n$ :$\nexists k \in \Z_{>0}: k < n: p \divides a_k$ where $a \divides b$ denotes that $a$ is a divisor of $b$.
Definition:Factorion
A '''factorion (base $b$)''' is a positive integer which, when expressed in base $b$, equals the sum of the factorials of its digits.
Definition:Philosophy
'''Philosophy''' is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. As there are considerable overlaps with mathematics, it is sometimes argued that mathematics can be considered a branch of '''philosophy'''. It is the controversial opinion of some mathematicians that philosophy is for people who are not smart enough for mathematics.
Definition:Polynomial Ring/Sequences
Let $R^{\left({\N}\right)}$ be the ring of sequences of finite support over $R$. Let $\iota : R \to R^{\left({\N}\right)}$ be the mapping defined as: :$\iota \left({r}\right) = \left \langle {r, 0, 0, \ldots}\right \rangle$. Let $X$ be the sequence $\left \langle {0, 1, 0, \ldots}\right \rangle$. The '''polynomial ring over $R$''' is the ordered triple $\left({R^{\left({\N}\right)}, \iota, X}\right)$.
Definition:Highly Abundant Number
Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''highly abundant''' : :$\forall m \in \Z_{>0}, m < n: \map \sigma m < \map \sigma n$ where $\map \sigma n$ is the $\sigma$ function of $n$.
Definition:Heptagonal Pyramidal Number
'''Heptagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a heptagonal pyramid.
Definition:Lychrel Number/Candidate
No natural number has been '''proved''' to be a '''Lychrel number''' as of time of writing (June $2017$).  However, plenty of numbers have not shown themselves to terminate in a palindromic number, although in some cases millions of iterations have been tested. Hence a '''candidate Lychrel number''' is a natural number which '''is not known to''' form a palindromic number through repeated iteration of the reverse-and add process.
Definition:Sequence/Minimizing/Functional
Let $y$ be a real mapping defined on a space $\MM$. {{Stub|what is $\MM$?}} Let $J \sqbrk y$ be a functional such that: :$\exists y \in \MM: J \sqbrk y < \infty$ :$\ds \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$ Let $\sequence {y_n}$ be a sequence such that: :$\ds \lim_{n \mathop \to \infty} J \sqbrk {y_n} = \mu$ Then the sequence $\sequence {y_n}$ is called a '''minimizing sequence (of the functional $J \sqbrk y$)'''.
Definition:Sequence/Minimizing/Functional/Limit Minimizing Function of
Let $\sequence {y_n}$ be a minimizing sequence of a functional $J$. Suppose: :$\displaystyle \lim_{n \mathop \to \infty} y_n = \hat y$ and :$\displaystyle \lim_{n \mathop \to \infty} J \sqbrk {y_n} = J \sqbrk {\hat y}$ Then $\hat y$ is the '''limit minimizing function''' of $J \sqbrk {y_n}$ and $J \sqbrk {\hat y} = \mu$.
Definition:Ritz Method
Let $\MM$ be a normed linear space. Let $J\sqbrk y$ be a functional defined on space $\MM$. Let $\sequence{\phi_n}$ be an infinite sequence of mappings in $\MM$. Let $\MM_n$ be an $n$-dimensional linear subspace of $\MM$, spanned by first $n$ mapping of $\sequence{\phi_n}$. Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$, where $\boldsymbol \alpha$ is a real $n$-dimensional vector. Minimise $J \sqbrk {\eta_n}$  $\boldsymbol\alpha$. Then $J \sqbrk {\eta_n}$ is an approximate minimum of $J \sqbrk y$, and is denoted by $\mu_n$. This method is known as '''Ritz method'''.
Definition:Multiplicative Magic Square
A '''multiplicative magic square''' is an arrangement of $n^2$ distinct numbers into an $n \times n$ square array such that: :the product of the elements of each row :the product of the elements in each column :the product of the elements along each diagonal are the same.
Definition:Multiplicative Magic Square/Order
An $n \times n$ multiplicative magic square is called an '''order $n$ multiplicative magic square'''.
Definition:Complete Ritz Sequence
Let $\MM$ be a normed linear space. Let $\sequence {\phi_n}$ be a Ritz sequence in $\MM$. Let $\MM_n$ be an $n$-dimensional linear subspace of $\MM$, spanned by the first $n$ mappings of $\sequence {\phi_n}$. Let $\eta_n$ be of the form: :$\eta_n = \boldsymbol \alpha \boldsymbol \phi$ where $\boldsymbol \alpha$ is an $n$-dimensional real vector. Suppose: :$\forall y \in \MM: \forall \epsilon > 0: \exists \map n \epsilon \in \N: \exists \eta_n \in \MM_n: \size {\eta_n - y} < \epsilon$ {{explain|which norm is it?}} Then the sequence $\sequence{\phi_n}$ is called '''complete''' in $\MM$. {{NamedforDef|Walther Ritz|cat = Ritz}}
Definition:Ritz Sequence
Let $\sequence {\phi_n}$ be an infinite sequence of mappings in a normed linear space, where: :$\phi_n: \R \to \R$ Let this sequence be constrained by the requirements defined in the definition of the Ritz method. {{explain|Specify what these are. For ultimate reusability, consider extracting those requirements into an appropriately-named and transcluded subpage of the above.}} Then the sequence $\set {\phi_n}$ can be called a '''Ritz Sequence'''.
Definition:Plato's Geometrical Number
The actual value of what is generally known as '''Plato's geometrical number''' is uncertain. The passage in question from {{AuthorRef|Plato}}'s {{BookLink|Republic|Plato}} is obscure and difficult to interpret: :''But the number of a human creature is the first number in which root and square increases, having received three distances and four limits, of elements that make both like and unlike wax and wane, render all things conversable and rational with one another.'' There are two numbers which it is generally believed it could be: :$216$ :$12 \, 960, \, 000$ It is believed that the expression: :''three distances and four limits'' refers to cubing. It is further believed that the reference is to the area of the $3-4-5$ triangle, which is $6$. The passage is also deduced to contain a reference to $2 \times 3$. It is also interpreted by other commentators as being $12 \, 960 \, 000$, which is $60^4$.
Definition:Fedorov Group
A '''Fedorov group''' is the symmetry group of a $3$-dimensional configuration in space.
Definition:Space Group
A '''space group''' is the symmetry group of a configuration in $n$-dimensional space.
Definition:Wallpaper Group
A '''wallpaper group''' is the symmetry group of a $2$-dimensional configuration in the plane.
Definition:Amicable Pair/Definition 1
$m$ and $n$ are an '''amicable pair''' : :the aliquot sum of $m$ is equal to $n$ and: :the aliquot sum of $n$ is equal to $m$.
Definition:Amicable Pair/Definition 2
$m$ and $n$ are an '''amicable pair''' : :$\map \sigma m = \map \sigma n = m + n$ where $\map \sigma m$ denotes the $\sigma$ function.
Definition:Amicable Pair/Definition 3
$m$ and $n$ are an '''amicable pair'''  they form a sociable chain of order $2$.
Definition:Thabit Pair
A '''Thabit pair''' is an amicable pair formed by an application of Thabit's Rule: :$\tuple {2^n a b, 2^n c}$ where $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime.
Definition:Thabit Number
A '''Thabit number''' is an integer in the form: :$3 \times 2^n - 1$ where $n \in \Z_{\ge 0}$ is a non-negative integer.
Definition:Thabit Number/Sequence
The sequence of Thabit numbers begins: :$2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, \ldots$
Definition:Thabit Prime
A '''Thabit prime''' is a Thabit number which is prime.
Definition:Thabit Prime/Sequence
The sequence of Thabit primes begins: :$2, 5, 11, 23, 47, 191, 383, 6143, 786 \, 431, 51 \, 539 \, 607 \, 551, \ldots$
Definition:Amicable Triplet
Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Triplet/Definition 1}} === Definition 2 === {{:Definition:Amicable Triplet/Definition 2}}
Definition:Amicable Triplet/Definition 1
$\tuple {m_1, m_2, m_3}$ are an '''amicable triplet'''  the aliquot sum of any one of them equals the sum of the other two: :the aliquot sum of $m_1$ is equal to $m_2 + m_3$ and: :the aliquot sum of $m_2$ is equal to $m_1 + m_3$ and: :the aliquot sum of $m_3$ is equal to $m_1 + m_2$
Definition:Amicable Triplet/Definition 2
$\left({m_1, m_2, m_3}\right)$ are an '''amicable triplet''' : :$\sigma \left({m_1}\right) = \sigma \left({m_2}\right) = \sigma \left({m_3}\right) = m_1 + m_2 + m_3$ where $\sigma \left({m}\right)$ denotes the $\sigma$ function.
Definition:Ljunggren Equation
The '''Ljunggren equation''' is the indeterminate Diophantine equation: :$x^2 + 1 = 2 y^4$
Definition:Sierpiński Number
There are two kinds of '''Sierpiński number''': === Sierpiński Numbers of the First Kind === {{:Definition:Sierpiński Number of the First Kind}} === Sierpiński Numbers of the Second Kind === {{:Definition:Sierpiński Number of the Second Kind}}
Definition:Sierpiński Number of the First Kind
The '''Sierpiński numbers of the first kind''' are the integers $S_n$ in the form: :$S_n := n^n + 1$ for all integers $n$.
Definition:Sierpiński Number of the Second Kind/Sequence
The  sequence of known '''Sierpiński numbers of the second kind''' starts: :$78\ 557, \ 271\ 129, \ 271\ 577, \ 322\ 523, \ 327\ 739, \ 482\ 719, \ 575\ 041, \ 603\ 713, \ 903\ 983, \ 934\ 909, \ 965\ 431, \ \ldots$
Definition:Sierpiński Number of the Second Kind/Historical Note
{{AuthorRef|Wacław Franciszek Sierpiński}} proved in $1960$ that there is an infinite number of Sierpiński numbers of the second kind.
Definition:Balanced Prime
=== Definition 1 === {{:Definition:Balanced Prime/Definition 1}} === Definition 2 === {{:Definition:Balanced Prime/Definition 2}} === Definition 3 === {{:Definition:Balanced Prime/Definition 3}}
Definition:Balanced Prime/Definition 2
Let $\paren {p_{n - 1}, p_n, p_{n + 1} }$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' : {{begin-eqn}} {{eqn | l = p_{n - 1} + d       | r = p_n       | c =  }} {{eqn | l = p_{n - 1} + 2 d       | r = p_{n + 1}       | c =  }} {{end-eqn}} for some $d \in \Z$.
Definition:Balanced Prime/Definition 1
Let $\left({p_{n - 1}, p_n, p_{n + 1} }\right)$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' : :$p_n = \dfrac {p_{n - 1} + p_{n + 1} } 2$
Definition:Aliquot Sequence
Let $m$ be a positive integer. Let $\map s m$ be the aliquot sum of $m$. Define the sequence $\sequence {a_k}$ recursively as: :$a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$ The sequence $\sequence {a_k}$ is known as an '''aliquot sequence'''.
Definition:Eighth Power
A '''eighth power''' is an integer which can be expressed as the $8$th power of an integer.
Definition:Superfactorial
The '''superfactorial of $n$''' is defined as: :$n\$ = \displaystyle \prod_{k \mathop = 1}^n k! = 1! \times 2! \times \cdots \times \left({n - 1}\right)! \times n!$ where $k!$ denotes the factorial of $n$.
Definition:Completely Irreducible
Let $\struct {S, \preceq}$ be an ordered set. Let $p \in S$. An element $p$ is '''completely irreducible'''  :$p^\succeq \setminus \set p$ admits a minimum element where $p^\succeq$ denotes the upper closure of $p$.
Definition:Poulet Number
Let $q$ be a composite number such that $2^q \equiv 2 \pmod q$. Then $q$ is a '''Poulet number'''.
Definition:Time/Unit/Lunar Month
The '''lunar month''' is defined as the length of time between $2$ new moons. :$1$ '''lunar month''' $\approx 29 \cdotp 530588$ days or: :$1$ '''lunar month''' $\approx 29$ days, $12$ hours, $44$ minutes and $2 \cdotp 8$ seconds.
Definition:Calendar
A '''calendar''' is a technique to keep track of the passing of time throughout the course of a year. Its objective is to provide a more-or-less standard way of identifying the time of year by associating a name and number to each day in the year.
Definition:Calendar/Julian/Year
A '''Julian year''' is the length of a year as defined using the '''Julian calendar'''. :$1$ '''Julian year''' $= \begin{cases} 366 \, \text {days} & : 4 \divides y \\ 365 \, \text {days} & : 4 \nmid y \end{cases}$ where: :$y$ denotes the number of the year :$4 \divides y$ denotes that $y$ is divisible by $4$ :$4 \nmid y$ denotes that $y$ is not divisible by $4$.
Definition:Calendar/Julian
The '''Julian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Julius_Caesar Julius Caesar] in $45 \, \text{BCE}$. It divides the year into: :$365$ days for every $3$ out of $4$ years and: :$366$ days for every $1$ out of $4$ years, known as a leap year. The years themselves are given an index number, and are known by that number. A standard numbering system was introduced by [https://en.wikipedia.org/wiki/Dionysius_Exiguus Dionysus Exiguus]. He identified a particular year as being $525 \, \text{A.D.}$, where $\text{A.D.}$ is an abbreviation for '''Anno Domini''', Latin for '''in the year of the Lord'''. The year $1$ was conventionally supposed to identify the year of the birth of [https://en.wikipedia.org/wiki/Jesus Jesus of Nazareth], although the accuracy of this has since been questioned. Years before $1 \, \text{A.D.}$ are counted backwards and assigned the label $\text{B.C.}$. However, note that the year immediately prior to $1 \, \text{A.D.}$ is $1 \, \text{B.C.}$, not the intuitive '''year $0$''', a discrepancy that can cause confusion. Using this system of numbering, a leap year is identified by this number being divisible by $4$. The $365$ or $366$ days in the year are divided into $12$ approximately equal sections called months, which are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = January       | cc= $31$ days }} {{eqn | l = 2:       | o =        | c = February       | cc= $28$ days, or $29$ days in a leap year }} {{eqn | l = 3:       | o =        | c = March       | cc= $31$ days }} {{eqn | l = 4:       | o =        | c = April       | cc= $30$ days }} {{eqn | l = 5:       | o =        | c = May       | cc= $31$ days }} {{eqn | l = 6:       | o =        | c = June       | cc= $30$ days }} {{eqn | l = 7:       | o =        | c = July       | cc= $31$ days }} {{eqn | l = 8:       | o =        | c = August       | cc= $31$ days }} {{eqn | l = 9:       | o =        | c = September       | cc= $30$ days }} {{eqn | l = 10:       | o =        | c = October       | cc= $31$ days }} {{eqn | l = 11:       | o =        | c = November       | cc= $30$ days }} {{eqn | l = 12:       | o =        | c = December       | cc= $31$ days }} {{end-eqn}} Thus, for example, the day following the $31$st of January is the $1$st of February, and the $30$th of June is followed by the $1$st of July.
Definition:Leap Year
The number of days in a year is not an integer, but lies somewhere between $365$ and $366$. However, it is convenient to assign an integer number of days to any given year. It is also desirable to ensure that a particular day of the year occurs at approximately the same time each year. In the Julian and Gregorian calendars: :some years are assigned $365$ days and: :some years are assigned $366$ days. The years which have $366$ days are called '''leap years'''.