Inmathematics, asquare number orperfect square is aninteger that is thesquare of an integer;[1] in other words, it is theproduct of some integer with itself. For example, 9 is a square number, since it equals32 and can be written as3 × 3.
The usual notation for the square of a numbern is not the productn × n, but the equivalentexponentiationn2, usually pronounced as "n squared". The namesquare number comes from the name of the shape. The unit ofarea is defined as the area of aunit square (1 × 1). Hence, a square with side lengthn has arean2. If a square number is represented byn 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 ofn; thus, square numbers are a type offigurate numbers (other examples beingcube numbers andtriangular numbers).
In thereal number system, square numbers arenon-negative. A non-negative integer is a square number when itssquare root is again an integer. For example, so 9 is a square number.
A positive integer that has no squaredivisors except 1 is calledsquare-free.
For a non-negative integern, thenth square number isn2, with02 = 0 being thezeroth one. The concept of square can be extended to some other number systems. Ifrational 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,.
Starting with 1, there are square numbers up to and includingm, where the expression represents thefloor of the number x.
The squares (sequenceA000290 in theOEIS) 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 identityn2 − (n − 1)2 = 2n − 1. 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,n2 = (n − 1)2 + (n − 1) +n.
The numberm is a square number if and only if one can arrangem points in a square:
m = 12 = 1
m = 22 = 4
m = 32 = 9
m = 42 = 16
m = 52 = 25
The expression for thenth square number isn2. This is also equal to the sum of the firstnodd 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:For example,52 = 25 = 1 + 3 + 5 + 7 + 9.
The sum of the firstn odd integers isn2.1 + 3 + 5 + ... + (2n − 1) =n2. Animated 3D visualization on a tetrahedron.
There are severalrecursive methods for computing square numbers. For example, thenth square number can be computed from the previous square byn2 = (n − 1)2 + (n − 1) + n = (n − 1)2 + (2n − 1). Alternatively, thenth 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, becausen2 = 2(n − 1)2 − (n − 2)2 + 2. For example,
The square minus one of a numberm is always the product of and that is,For example, since72 = 49, one has. Since aprime number has factors of only1 and itself, and sincem = 2 is the only non-zero value ofm to give a factor of1 on the right side of the equation above, it follows that3 is the only prime number one less than a square (3 = 22 − 1).
More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is,This is thedifference-of-squares formula, which can be useful for mental arithmetic: for example,47 × 53 can be easily computed as502 − 32 = 2500 − 9 = 2491.A square number is also the sum of two consecutivetriangular numbers. The sum of two consecutive square numbers is acentered square number. Every odd square is also acentered 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 aneven 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 form4k(8m + 7). A positive integer can be represented as a sum of two squares precisely if itsprime factorization contains no odd powers of primes of the form4k + 3. This is generalized byWaring's problem.
Inbase 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.
Inbase 12, a square number can end only with square digits (like in base 12, aprime 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).[citation needed] All such rules can be proved by checking a fixed number of cases and usingmodular arithmetic.
In general, if aprimep divides a square number m then the square ofp must also dividem; ifp fails to dividem/p, thenm 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 numberm is a square number if and only if, in itscanonical representation, all exponents are even.
Squarity testing can be used as alternative way infactorization of large numbers. Instead of testing for divisibility, test for squarity: for givenm and some number k, ifk2 −m is the square of an integer n thenk −n dividesm. (This is an application of the factorization of adifference of two squares.) For example,1002 − 9991 is the square of 3, so consequently100 − 3 divides 9991. This test is deterministic for odd divisors in the range fromk −n tok +n wherek covers some range of natural numbers
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 explainsGalileo'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, foru = 0 and constanta (acceleration due to gravity without air resistance); sos is proportional tot2, and the distance from the starting point are consecutive squares for integer values of time elapsed.[2]
The sum of then firstcubes is the square of the sum of then first positive integers; this isNicomachus's theorem.
All fourth powers, sixth powers, eighth powers and so on are perfect squares.
Squares of even numbers are even, and are divisible by 4, since(2n)2 = 4n2. Squares of odd numbers are odd, and are congruent to 1modulo 8, since(2n + 1)2 = 4n(n + 1) + 1, andn(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 acentered 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 of2n differ by an amount containing an odd factor, the only perfect square of the form2n − 1 is 1, and the only perfect square of the form2n + 1 is 9.
If the number is of the formm5 wherem represents the preceding digits, its square isn25 wheren =m(m + 1) and represents digits before 25. For example, the square of 65 can be calculated byn = 6 × (6 + 1) = 42 which makes the square equal to 4225.
If the number is of the formm0 wherem represents the preceding digits, its square isn00 wheren =m2. For example, the square of 70 is 4900.
If the number has two digits and is of the form5m wherem represents the units digit, its square isaabb whereaa = 25 +m andbb =m2. For example, to calculate the square of 57,m = 7 and25 + 7 = 32 and72 = 49, so572 = 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 in376. (The numbers 5, 6, 25, 76, etc. are calledautomorphic numbers. They are sequenceA003226 in theOEIS.[3])
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 25,242 = 576 and262 = 676, both ending in 76. In general,. 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).
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