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In mathematics, a square number or perfect square is an integer that is the square of an integer;[1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3.

The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "Template:Mvar 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 (1 × 1). Hence, a square with side length Template:Mvar has area n2.

Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number is that its square root is again an integer. For example, $ \sqrt{9}=3 $, so 9 is a square number.

A positive integer that has no perfect square divisors except 1 is called square-free.

For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, e.g., $ \textstyle \frac{4}{9} = \left(\frac{2}{3}\right)^2 $.

Starting with 1, there are ⌊$ \sqrt{m} $⌋ square numbers up to and including m, where the expression ⌊x⌋ represents the floor of the number x.

TableEdit

The following is list of numbers up to 100 and their squares.

All square numbers end with square digits (0, 1, 4 or 9).

nn2nn2nn2nn2
1131961613101916X61
2432X04623204927004
3933X69633309937169
41434E14643414947314
52135E81653521957481
630361030663630967630
7413710X16737419777X1
854381154683854987954
969391209693969997E09
X843X12846X3X849X8084
EX13E13416E3EX19E8241
10100401400704100X08400
11121411481714221X18581
12144421544724344X28744
13169431609734469X38909
14194441694744594X48X94
15201451761754701X59061
16230461830764830X69230
17261471901774961X79401
18294481994784X94X89594
19309491X69795009X99769
1X3444X1E447X5144XX9944
1E3814E20217E5281XE9E21
20400502100805400E0X100
214415121X1815541E1X2X1
22484522284825684E2X484
23509532369835809E3X669
24554542454845954E4X854
255X1552541855XX1E5XX41
26630562630866030E6E030
27681572721876181E7E221
28714582814886314E8E414
29769592909896469E9E609
2X8045X2X048X6604EXE804
2E8615E2E018E6761EEEX01
3090060300090690010010000

PropertiesEdit

The number m is a square number if and only if one can arrange m points in a square:

The expression for the Template:Mvarth square number is Template:Math. This is also equal to the sum of the first Template:Mvar 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_{k=1}^n(2k-1). $

So for example, 52 = 21 = 1 + 3 + 5 + 7 + 9.

There are several recursive methods for computing square numbers. For example, the nth square number can be computed from the previous square by n2 = (n − 1)2 + (n − 1) + n = (n − 1)2 + (2n − 1). Alternatively, the Template:Mvarth square number can be calculated from the previous two by doubling the Template:Mathth square, subtracting the Template:Mathth square number, and adding 2, because n2 = 2(n − 1)2 − (n − 2)2 + 2. For example,

2 × 52 − 42 + 2 = 2 × 21 − 14 + 2 = 42 − 14 + 2 = 30 = 62

One number less than a square (m - 1) is always the product of

$ \sqrt{m} $ - 1 and

$ \sqrt{m} $ + 1 (e.g. 8 × 6 equals 40, while 72 equals 41). Thus, 3 is the only prime number one less than a square.

A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.

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 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 Template:Math. 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 Template:Math. This is generalized by Waring's problem.

A square number can end only with square digits (like the case of the prime numbers, a prime number can end only with prime digits or 1), i.e. 0, 1, 4 or 9, as follows:

  • if a number is divisible both by 2 and by 3 (i.e. divisible by 6), its square ends in 0, and only 0, 3 can precede it (also, the number form by the preceding digits, divided by 3, is also a square number) (if ending with 00, then the preceding digits must also form a square number, and if ending with 30, then the preceding digits must form a Pronic number);
  • if a number is divisible neither by 2 nor by 3, its square ends in 1, and only even numbers can precede it (i.e. the number form by the preceding digits must be even) (also, the number form by the preceding digits, divided by 2, is a generalized pentagonal number);
  • if a number is divisible by 2, but not by 3, its square ends in 4, and only 0, 1, 4, 5, 8, 9 can precede it (i.e. the number form by the preceding digits must be = 0 or 1 mod 4) (also, the number form by the preceding digits is a generalized octagonal number);
  • if a number is not divisible by 2, but by 3, its square ends in 9, and only 0, 6 can precede it (i.e. the number form by the preceding digits must be divisible by 6) (also, the number form by the preceding digits, divided by 6, is a triangular number, thus the number form by the preceding digits, added by 1, is a centered hexagonal number (or a hex number)).

Also....

  • If we write a square number, and write digits "00" after this number, then we get a square number.
  • If we write a Pronic number, and write digits "30" after this number, then we get a square number.
  • If we write a number which is 3 times a square number, and write a digit "0" after this number, then we get a square number.
  • If we write a number which is twice a generalized pentagonal number, and write a digit "1" after this number, then we get a square number.
  • If we write a generalized octagonal number, and write a digit "4" after this number, then we get a square number.
  • If we write a number which is 6 times a triangular number, and write a digit "9" after this number, then we get a square number.

Similar rules can be given for other bases, or for earlier digits (the dozens digit 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 Template:Mvar divides a square number Template:Mvar then the square of Template:Mvar must also divide Template:Mvar; if Template:Mvar fails to divide Template:Math, then Template:Mvar 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 Template:Mvar 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 of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if

$ k^2-m $ is the square of an integer n then kn divides m. (This is an application of the factorization of a difference of two squares.) For example, 1002 − EEE3 is the square of 3, so consequently 100 − 3 divides EEE3. This test is deterministic for odd divisors in the range from kn to k + n where k covers some range of natural numbers

$ k \ge \sqrt{m} $ .

A square number cannot be a perfect number.

The sum of the n first square numbers is

$ \sum_{n=0}^N n^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = \frac{N(N+1)(2N+1)}{6}. $

The first values of these sums, the square pyramidal numbers, are:

0, 1, 5, 12, 26, 47, 77, E8, 150, 1E9, 281, 362, 462, 583, 707, 874, X48, 1049, 1279, 151X, 17E2, 1XEE, 2243, 2604, 2X04

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.

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.

Special casesEdit

  • If the number is of the form m6 where m represents the preceding digits, its square is n30 where n = m(m + 1) and represents digits before 30. For example, the square of 86 can be calculated by n = 8 × (8 + 1) = 60 which makes the square equal to 6030.
  • If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m2. For example, the square of 70 is 4100.
  • If the number has two digits and is of the form 6m where m represents the units digit, its square is aabb where aa = 30 + m and bb = m2. Example: To calculate the square of 68, 30 + 8 = 38 and 82 = 54, which means 682 = 3854.
  • If the number ends in 4, its square will end in 4; similarly for ending in 54, 854, 3854, ..., 2E21E61E3854, etc. If the number ends in 9, its square will end in 9, similarly for ending in 69, 369, 8369, ..., 909X05X08369. For example, the square of 75369 is 4745E22369, both ending in 369. (The numbers 4, 9, 54, 69, etc. are called automorphic numbers)

Except 0, 1 and 100, square numbers cannot be Fibonacci numbers. (note that these three numbers except 0 are exactly the only two Lucas numbers which are square numbers written in binary (base 2))

Except 1 and 4, square numbers cannot be Lucas numbers.

Except 1, square numbers cannot be repunits.

Except single-digit square numbers (0, 1, 4 and 9), square numbers cannot be repdigits.

Except 0, square numbers cannot be pronic numbers.

Except 0, 1 and 2X04, square numbers cannot be square pyramidal numbers.

Note that both of 21 and 201 are square numbers, are there any other square numbers of the form 2{0}1 (=2000...0001)?

Odd and even square numbersEdit

Squares of even numbers are even (and in fact divisible by 4), since (2n)2 = 4n2.

Squares of odd numbers are odd, since (2n + 1)2 = 4(n2 + n) + 1.

It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

As all even square numbers are divisible by 4 (in fact, end with 0 or 4, a square cannot end with 8), the even numbers of the form 4n + 2 (i.e. end with 2, 6 or X) are not square numbers.

As all odd square numbers are of the form 4n + 1 (in fact, end with 1 or 9, a square cannot end with 5), the odd numbers of the form 4n + 3 (i.e. end with 3, 7 or E) are not square numbers.

All odd square numbers are of the form 8n + 1, since (2n + 1)2 = 4n(n + 1) + 1 and n(n + 1) is an even number. (thus, the dozens digit of an odd square number is even, this is because the units digit of an odd square number is either 1 or 9)

Every odd perfect square is a centered octagonal number (centered 8-gonal number). 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 2n differ by an amount containing an odd factor, the only perfect square of the form 2n − 1 is 1, and the only perfect square of the form 2n + 1 is 9.


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