Square number

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File:Square number 16 as sum of gnomons.svg

Square number 16 as sum of gnomons.

In mathematics, a square number or perfect square is an integer that is the square of an integer;<ref>Some authors also call squares of rational numbers perfect squares.</ref> in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals Template:Math and can be written as Template:Math.

The usual notation for the square of a number Template:Mvar is not the product Template:Math, but the equivalent exponentiation Template:Math, 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 (Template:Math). Hence, a square with side length Template:Mvar has area Template:Math. 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).

In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, <math>\sqrt{9} = 3,</math> so 9 is a square number.

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

For a non-negative integer Template:Mvar, the Template:Mvarth square number is Template:Math, with Template:Math 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, for example, <math>\textstyle \frac{4}{9} = \left(\frac{2}{3}\right)^2</math>.

Starting with 1, there are <math>\lfloor \sqrt{m} \rfloor</math> square numbers up to and including Template:Mvar, where the expression <math>\lfloor x \rfloor</math> represents the floor of the number Template:Mvar.

ExamplesEdit

The squares (sequence A000290 in the OEIS) 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 Template:Math. 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, Template:Math.

PropertiesEdit

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

Template:Bigmath	File:Square number 1.png
Template:Bigmath	File:Square number 4.png
Template:Bigmath	File:Square number 9.png
Template:Bigmath	File:Square number 16.png
Template:Bigmath	File:Square number 25.png

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:<math display="block">n^2 = \sum_{k=1}^n (2k-1).</math>For example, Template:Math.

File:The sum of the first n odd integers is n². 1+3+5+...+(2n-1)=n²..gif

The sum of the first n odd integers is n². Template:Nowrap. Animated 3D visualization on a tetrahedron.

There are several recursive methods for computing square numbers. For example, the Template:Mvarth square number can be computed from the previous square by Template:Math. 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 Template:Math. For example,

Template:Math.

The square minus one of a number Template:Mvar is always the product of <math>m - 1</math> and <math>m + 1;</math> that is,<math display="block">m^2-1=(m-1)(m+1).</math>For example, since Template:Math, one has <math>6 \times 8 = 48</math>. Since a prime number has factors of only Template:Math and itself, and since Template:Math is the only non-zero value of Template:Math to give a factor of Template:Math on the right side of the equation above, it follows that Template:Math is the only prime number one less than a square (Template:Math).

More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is,<math display="block">a^2-b^2=(a+b)(a-b)</math>This is the difference-of-squares formula, which can be useful for mental arithmetic: for example, Template:Math can be easily computed as Template:Math. 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.

In base 10, 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 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, a square number can end only with square digits (like in base 12, a prime number 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, and its preceding digit must be 0 or 3;
if a number is divisible neither by 2 nor by 3, its square ends in 1, and its preceding digit must be even;
if a number is divisible by 2, but not by 3, its square ends in 4, and its preceding digit must be 0, 1, 4, 5, 8, or 9; and
if a number is not divisible by 2, but by 3, its square ends in 9, and its preceding digit must be 0 or 6.

Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example).Template:Citation needed 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 Template:Mvar and some number Template:Mvar, if Template:Math is the square of an integer Template:Mvar then Template:Math divides Template:Mvar. (This is an application of the factorization of a difference of two squares.) For example, Template:Math is the square of 3, so consequently Template:Math divides 9991. This test is deterministic for odd divisors in the range from Template:Math to Template:Math where Template:Mvar covers some range of natural numbers <math>k \geq \sqrt{m}.</math>

A square number cannot be a perfect number.

The sum of the n first square numbers is<math display="block">\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}.</math>The first values of these sums, the square pyramidal numbers, are: (sequence A000330 in the OEIS)

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...

File:Proofwithoutwords.svg

Proof without words for the sum of odd numbers theorem

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: 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 <math>s=ut+\tfrac{1}{2}at^2</math>, for Template:Math and constant Template:Math (acceleration due to gravity without air resistance); so Template:Math is proportional to Template:Math, and the distance from the starting point are consecutive squares for integer values of time elapsed.<ref>Template:Cite book</ref>

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 <math>T_n</math> is:<math display="block">(T_n)^2 + (T_{n+1})^2 = T_{(n+1)^2}</math>

Odd and even square numbersEdit

File:Visual proof centered octagonal numbers are odd squares.svg

Proof without words that all centered octagonal numbers are odd squares

Squares of even numbers are even, and are divisible by 4, since Template:Math. Squares of odd numbers are odd, and are congruent to 1 modulo 8, since Template:Math, and Template:Math 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. 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 Template:Math differ by an amount containing an odd factor, the only perfect square of the form Template:Math is 1, and the only perfect square of the form Template:Math is 9.

Special casesEdit

If the number is of the form Template:Math where Template:Math represents the preceding digits, its square is Template:Math where Template:Math and represents digits before 25. For example, the square of 65 can be calculated by Template:Math which makes the square equal to 4225.
If the number is of the form Template:Math where Template:Math represents the preceding digits, its square is Template:Math where Template:Math. For example, the square of 70 is 4900.
If the number has two digits and is of the form Template:Math where Template:Math represents the units digit, its square is Template:Math where Template:Math and Template:Math. For example, to calculate the square of 57, Template:Math and Template:Math and Template:Math, so Template:Math.
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 numbers. They are sequence A003226 in the OEIS.<ref>Template:Cite OEIS</ref>)
In base 10, the last two digits of square numbers follow a repeating pattern mirrored symmetrical around multiples of 25. In the example of 24 and 26, both 1 off from 25, Template:Math and Template:Math, both ending in 76. In general, <math display="inline">(25n+x)^2-(25n-x)^2=100nx</math>. 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).

NotesEdit

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Square number

Contents

ExamplesEdit

PropertiesEdit

Odd and even square numbersEdit

Special casesEdit

See alsoEdit

NotesEdit

Further readingEdit