Square number

A square number, sometimes also called a perfect square, is the result of an integer multiplied by itself. 1, 4, 9, 16 and 25 are the first five square numbers.
In a formula, the square of a number n is denoted  (exponentiation), usually pronounced as "n squared".  The name square number comes from the name of the shape; see below.

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,  = 3, so 9 is a square number.

Examples 
The squares  smaller than 702 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

 
602 = 3600
612 = 3721 
622 = 3844
632 = 3969
642 = 4096
652 = 4225
662 = 4356
672 = 4489
682 = 4624
692 = 4761

There are infinitely many square numbers, as there are infinitely many natural numbers.

Properties
The number m is a square number if and only if one can compose a square of m equal (lesser) squares:

A square with side length n has area .

The expression for the nth square number is n2. This is also equal to the sum of the first n 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:

So for example, .

A square number can end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows:
If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square.  
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.
If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
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.
If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
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.

All fourth powers, sixth powers, eighth powers and so on are perfect squares.

Special cases
 If the number is of the form m5 where m represents the preceding digits, its square is n25 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 m0 where m represents the preceding digits, its square is n00 where . For example the square of 70 is 4900.
 If the number has two digits and is of the form 5m where m represents the units digit, its square is AABB where  and . Example: To calculate the square of 57, 25 + 7 = 32 and 72 = 49, which means 572 = 3249.

Odd and even square numbers

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

Squares of odd numbers are odd, since .

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, the even numbers of the form  are not square numbers.

As all odd square numbers are of the form , the odd numbers of the form  are not square numbers.

Squares of odd numbers are of the form , since  and  is an even number.

Notes

References

Further reading 
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30–32, 1996.

Other websites 
 Learn Square Numbers . Practice square numbers up to 144 with this children's multiplication game
 Dario Alpern, Sum of squares. A Java applet to decompose a natural number into a sum of up to four squares.
Fibonacci and Square Numbers  at Convergence 
 The first 1,000,000 perfect squares Includes a program for generating perfect squares up to 1015.

 
Number theory
Integer sequences