# Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.

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

The usual notation for the formula for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "n squared". For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth square. 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., 4/9 = (2/3)2).

Starting with 1, there are ${\displaystyle \lfloor {\sqrt {m}}\rfloor }$ square numbers up to and including m.

## Examples

The first 50 squares of natural numbers (sequence A000290 in the OEIS) are:

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

The pattern between any perfect square from negative infinity to positive infinity is as follows,

${\displaystyle n^{2}=(n-1)^{2}+(2n-1).\ }$

## Properties

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

 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25

The formula for the nth square number is n2. This is also equal to the sum of the first n odd numbers

${\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1)}$

as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (marked as '+'). So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.

The nth square number can be calculated from the previous two by doubling the (n − 1)-th square, subtracting the (n − 2)-th square number, and adding 2, because n2 = 2(n − 1)2 − (n − 2)2 + 2. For example, 2 × 52 − 42 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 62.

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.

Lagrange's four-square theorem states that any positive integer can be written as the sum of 4 or fewer perfect squares. Three squares are not sufficient for numbers of the form 4k(8m + 7). 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 4k + 3. This is generalized by Waring's problem.

A square number can only end with digits 00,1,4,6,9, or 25 in base 10, as follows:

1. If the last digit of a number is 0, its square ends in 00 and the preceding digits must also form a square.
2. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
3. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
4. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
5. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
6. If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

A square number cannot be a perfect number.

## Easy ways to calculate square numbers

The formula to calculate any square number is: (n1)2 + (dn1) + (dn2)
An easy way to find square numbers is to find two numbers which have a mean of it, 212:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22 × 20 = 440 and 440 + 12 = 441. This works because of the identity

${\displaystyle (x-y)(x+y)=x^{2}-y^{2}\ }$

known as the difference of two squares. Thus (21 − 1)(21 + 1) = 212 − 12 = 440, if you work backwards.

#### Special cases

• If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m × (m + 1) and represents digits before 25. For example the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225.
• 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 4900.

## Odd and even square numbers

Squares of even numbers are even, 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.

## Chen's theorem

Chen Jingrun showed in 1975 that there always exists a number P which is either a prime or product of two primes between n2 and (n + 1)2. See also Legendre's conjecture.