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In mathematics, a square number or perfect square is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
that is the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of an integer; in other words, it is the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being cube numbers and
triangular numbers A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
). Square numbers are
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
. A non-negative integer is a square number when its square root is again an integer. For example, $\sqrt = 3,$ so 9 is a square number. A positive integer that has no square
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s except 1 is called
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
. For a non-negative integer , the th square number is , with being the
zeroth 0th or zeroth may refer to: Mathematics, science and technology * 0th or zeroth, an ordinal for the number zero * 0th dimension, a topological space * 0th element, of a data structure in computer science * Zeroth (software), deep learning softw ...
one. The concept of square can be extended to some other number systems. If
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example, $\textstyle \frac = \left\left(\frac\right\right)^2$. Starting with 1, there are $\lfloor \sqrt \rfloor$ square numbers up to and including , where the expression $\lfloor x \rfloor$ represents the floor of the number .

# Examples

The squares smaller than 602 = 3600 are:
:02 = 0 :12 = 1 :22 = 4 :32 = 9 :42 = 16 :52 = 25 :62 = 36 :72 = 49 :82 = 64 :92 = 81
:102 = 100 :112 = 121 :122 = 144 :132 = 169 :142 = 196 :152 = 225 :162 = 256 :172 = 289 :182 = 324 :192 = 361
:202 = 400 :212 = 441 :222 = 484 :232 = 529 :242 = 576 :252 = 625 :262 = 676 :272 = 729 :282 = 784 :292 = 841
:302 = 900 :312 = 961 :322 = 1024 :332 = 1089 :342 = 1156 :352 = 1225 :362 = 1296 :372 = 1369 :382 = 1444 :392 = 1521
:402 = 1600 :412 = 1681 :422 = 1764 :432 = 1849 :442 = 1936 :452 = 2025 :462 = 2116 :472 = 2209 :482 = 2304 :492 = 2401
:502 = 2500 :512 = 2601 :522 = 2704 :532 = 2809 :542 = 2916 :552 = 3025 :562 = 3136 :572 = 3249 :582 = 3364 :592 = 3481
The difference between any perfect square and its predecessor is given by the identity . Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, .

# Properties

The number ''m'' is a square number if and only if one can arrange ''m'' points in a square: The expression for the th square number is . This is also equal to the sum of the first odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows: :$n^2 = \sum_^n\left(2k-1\right).$ For example, . There are several
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
methods for computing square numbers. For example, the th square number can be computed from the previous square by . Alternatively, the th square number can be calculated from the previous two by doubling the th square, subtracting the th square number, and adding 2, because . For example, :. The square minus one of a number is always the product of $m - 1$ and $m + 1;$ that is, :$m^2-1=\left(m-1\right)\left(m+1\right).$ For example, since $7^2=49,$ one has $6\times 8=48.$ It follows that is the only
prime number A prime number (or a prime) 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 ...
one less than a square (). More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is, :$a^2-b^2=\left(a+b\right)\left(a-b\right)$ (this is the difference-of-squares formula). This can be useful for mental arithmetic: for example, can be easily computed as . A square number is also the sum of two consecutive
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s. The sum of two consecutive square numbers is a
centered square number In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each cen ...
. Every odd square is also a
centered octagonal number A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the od ...
. Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.
Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...
states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form . This is generalized by
Waring's problem 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 numb ...
. In
base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
, a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows: * if the last digit of a number is 0, its square ends in 0 (in fact, the last two digits must be 00); * if the last digit of a number is 1 or 9, its square ends in an even digit followed by a 1; * if the last digit of a number is 2 or 8, its square ends in an even digit followed by a 4; * if the last digit of a number is 3 or 7, its square ends in an even digit followed by a 9; * if the last digit of a number is 4 or 6, its square ends in an odd digit followed by a 6; and * if the last digit of a number is 5, its square ends in 25. In
base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...
, a square number can end only with square digits (like in base 12, a
prime number A prime number (or a prime) 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 ...
can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows: * if a number is divisible both by 2 and by 3 (that is, divisible by 6), its square ends in 0; * if a number is divisible neither by 2 nor by 3, its square ends in 1; * if a number is divisible by 2, but not by 3, its square ends in 4; and * if a number is not divisible by 2, but by 3, its square ends in 9. Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). All such rules can be proved by checking a fixed number of cases and using modular arithmetic. In general, if a prime  divides a square number  then the square of must also divide ; if fails to divide , then is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number is a square number if and only if, in its canonical representation, all exponents are even. Squarity testing can be used as alternative way in
factorization 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 ...
of large numbers. Instead of testing for divisibility, test for squarity: for given and some number , if is the square of an integer  then divides . (This is an application of the factorization of a
difference of two squares In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity :a^2-b^2 = (a+b)(a-b) in elementary algebra. P ...
.) For example, is the square of 3, so consequently divides 9991. This test is deterministic for odd divisors in the range from to where covers some range of natural numbers $k \geq \sqrt.$ A square number cannot be a
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...
. The sum of the ''n'' first square numbers is : $\sum_^N n^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = \frac.$ The first values of these sums, the square pyramidal numbers, are:
0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201...
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains
Galileo's law of odd numbers In classical mechanics and kinematics, Galileo's law of odd numbers states that the distance covered by a falling object in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers one ...
: if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length. From ''s'' = ''ut'' + ''at''2, for ''u'' = 0 and constant ''a'' (acceleration due to gravity without air resistance); so ''s'' is proportional to ''t''2, and the distance from the starting point are consecutive squares for integer values of time elapsed. The sum of the ''n'' first cubes is the square of the sum of the ''n'' first positive integers; this is Nicomachus's theorem. All fourth powers, sixth powers, eighth powers and so on are perfect squares. A unique relationship with triangular numbers $T_n$ is: :$\left(T_n\right)^2+\left(T_\right)^2=T_$

# Odd and even square numbers

Squares of even numbers are even, and are divisible by 4, since (2''n'')2 = 4''n''2. Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2''n'' + 1)2 = 4''n''(''n'' + 1) + 1, and ''n''(''n'' + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a
centered octagonal number A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the od ...
. The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of differ by an amount containing an odd factor, the only perfect square of the form is 1, and the only perfect square of the form is 9.

# Special cases

* If the number is of the form where represents the preceding digits, its square is where and represents digits before 25. For example, the square of 65 can be calculated by which makes the square equal to 4225. * If the number is of the form where represents the preceding digits, its square is where . For example, the square of 70 is 4900. * If the number has two digits and is of the form where represents the units digit, its square is where and . Example: To calculate the square of 57, 25 + 7 = 32 and 72 = 49, which means 572 = 3249. * If the number ends in 5, its square will end in 5; similarly for ending in 25, 625, 0625, 90625, ... 8212890625, etc. If the number ends in 6, its square will end in 6, similarly for ending in 76, 376, 9376, 09376, ... 1787109376. For example, the square of 55376 is 3066501376, both ending in ''376''. (The numbers 5, 6, 25, 76, etc. are called
automorphic number In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natura ...
s. They are sequence A003226 in the
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
.) * In base 10, the last two digits of square numbers follow a repeating pattern mirror symmetrical around multiples of 25, so for example, 242=576 and 262=676, and in general (25n+x)2-(25n-x)2=100nx. An analogous pattern applies for the last 3 digits around multiples of 250, and so on. As a consequence, of the 100 possible last 2 digits, only 22 of them occur among square numbers (since 00 and 25 are repeated).