A square number is a figurate number of the form , where n is an integer. A square number is also called a perfect square. The first few square numbers are 1, 4, 9, 16, 25, 36, 49, ... (Sloane's A000290).

A plot of the first few square numbers represented as a sequence of binary bits is shown above. The top portion shows to , and the bottom shows the next 510 values.

The generating function giving the square numbers is

The st square number is given in terms of the nth square number by

since

which is equivalent to adding a gnomon to the previous square, as illustrated above.

The nth square number is equal to the sum of the st and nth triangular numbers,

as can seen in the above diagram, in which the st triangular number is represented by the white triangles, the nth triangular number is represented by the black triangles, and the total number of triangles is the square number (R. Sobel, pers. comm.).

Square numbers can also be generated by taking the product of two consecutive even or odd numbers and adding 1. The result obtained by carrying out this operation is then the square of the average of the initial two numbers,

(E. K. Herrstrom, pers. comm.).

As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 4 positive squares (; Lagrange's four-square theorem), that every "sufficiently large" integer is a sum of no more than 4 positive squares (), and that every integer is a sum of at most 3 signed squares ( ). Actually, the basis set for representing positive integers with positive squares is , so 49 need never be used. Furthermore, since an infinite number of n require four squares to represent them, the least integer such that every positive integer beyond a certain point requires squares is given by .

The number of representation of a number n by k squares, distinguishing signs and order, is denoted and called the sum of squares function. The minimum number of squares needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 1, 2, 3, 4, 2, 1, 2, ... (Sloane's A002828), and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of squares are 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, ... (Sloane's A001156). A brute-force algorithm for enumerating the square partitions of n is repeated application of the greedy algorithm. However, this approach rapidly becomes impractical since the number of representations grows extremely rapidly with n, as shown in the following table.

The kth nonsquare number is given by

where is the floor function, and the first few are 2, 3, 5, 6, 7, 8, 10, 11, ... (Sloane's A000037).

The only numbers that are simultaneously square and pyramidal (the cannonball problem) are and , corresponding to and (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59; Ogilvy 1988), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). The cannonball problem is equivalent to solving the Diophantine equation

(Guy 1994, p. 147).

The only numbers that are square and tetrahedral are , , and (giving , , and ), as proved by Meyl (1878; cited in Dickson 1952, p. 25; Guy 1994, p. 147). In general, proving that only certain numbers are simultaneously figurate in two different ways is far from elementary.

To find the possible last digits for a square number, write for the number written in decimal notation as (a, b = 0, 1, ..., 9). Then

so the last digit of is the same as the last digit of . The following table gives the last digit of for b = 0, 1, ..., 9 (where numbers with more that one digit have only their last digit indicated, i.e., 16 becomes _6). As can be seen, the last digit can be only 0, 1, 4, 5, 6, or 9.

We can similarly examine the allowable last two digits by writing as

so

so the last two digits must have the last two digits of . Furthermore, the last two digits can be obtained by considering only b = 0, 1, 2, 3, and 4, since

has the same last two digits as (with the one additional possibility that c = 0 in which case the last two digits are 00). The following table (with the addition of 00) therefore exhausts all possible last two digits.

The only 22 possibilities are therefore 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, and 96, which can be summarized succinctly as 00, , , 25, , and , where e stands for an even number and o for an odd number. Additionally, a necessary (but not sufficient) condition for a number to be square is that its digital root be 1, 4, 7, or 9. The digital roots of the first few squares are 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, ... (Sloane's A056992), while the list of number having digital roots 1, 4, 7, or 9 is 1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, ... (Sloane's A056991).

The following table gives the possible residues mod n for square numbers for n = 1 to 20. The quantity s(n) gives the number of distinct residues for a given n.

In general, the odd squares are congruent to 1 (mod 8) (Conway and Guy 1996). Stangl (1996) gives an explicit formula by which the number of squares s(n) in (i.e., mod n) can be calculated. Let p be an odd prime. Then s(n) is the multiplicative function given by

s(n) is related to the number q(n) of quadratic residues in by

for (Stangl 1996).

For a perfect square n, or 1 for all odd primes p < n where is the Legendre symbol. A number n that is not a perfect square but that satisfies this relationship is called a pseudosquare.

In a Ramanujan conference talk, W. Gosper conjectured that every sum of four distinct odd squares is the sum of four distinct even squares. This conjecture was proved by M. Hirschhorn using the identity

	(18)

where a, b, c, and d are positive or negative integers. Hirschhorn also showed that every sum of four distinct oddly even squares is the sum of four distinct odd squares.

A prime number p can be written as the sum of two squares iff is not divisible by 4 the (Fermat's 4n plus 1 theorem). An arbitrary positive number n is expressible as the sum of two squares iff, given its prime factorization

none of is divisible by 4 (Conway and Guy 1996, p. 147). This is equivalent the requirement that all the odd factors of the squarefree part of n are equal to 1 (mod 4) (Hardy and Wright 1979, Finch). The first few numbers that can be expressed as the sum of two squares are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, ... (Sloane's A001481). Letting be the fraction of numbers that are expressible as the sum of two squares,

and

where K is the Landau-Ramanujan constant.

Numbers expressible as the sum of three squares are those not of the form for (Nagell 1951, p. 194; Wells 1986, pp. 48 and 56; Hardy 1999, p. 12).

The following table gives the first few numbers which require N = 1, 2, 3, and 4 squares to represent them as a sum (Wells 1986, p. 70).

Fermat's 4n plus 1 theorem guarantees that every prime of the form is a sum of two square numbers in only one way.

There are only 31 numbers that cannot be expressed as the sum of distinct squares: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128 (Sloane's A001422; Guy 1994; Savin 2000). The following numbers cannot be represented using fewer than five distinct squares: 55, 88, 103, 132, 172, 176, 192, 240, 268, 288, 304, 368, 384, 432, 448, 496, 512, and 752, together with all numbers obtained by multiplying these numbers by a power of 4. This gives all known such numbers less than 10⁵ (Savin 2000). All numbers can be expressed as the sum of at most five distinct squares, and only

and

require six distinct squares (Bohman et al. 1979; Guy 1994, p. 136; Savin 2000). In fact, 188 can also be represented using seven distinct squares:

The following table gives the numbers that can be represented in W different ways as a sum of S squares. For example,

can be represented in two ways (W = 2) by two squares (S = 2).

The least numbers that are the sum of two squares in exactly n different ways for n = 1, 2, ... are given by 2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, ... (Sloane's A016032; Beiler 1966, pp. 140-141; Rubin 1977-78; Culberson 1978-79; Hardy and Wright 1979; Rivera).

The product of four distinct nonzero integers in arithmetic progression is square only for (-3, -1, 1, 3), giving (Le Lionnais 1983, p. 53). It is possible to have three squares in arithmetic progression, but not four (Dickson 1952, pp. 435-440). If these numbers are , , and , there are positive integers p and q such that

where and one of r, s, or t is even (Dickson 1952, pp. 437-438). Every three-term progression of squares can be associated with a Pythagorean triple ) by
(Robertson 1996).

Catalan's conjecture states that 8 and 9 (2³ and 3²) are the only consecutive powers (excluding 0 and 1), i.e., the only solution to Catalan's Diophantine problem. This conjecture has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the conjecture not hold. It is also known that 8 and 9 are the only consecutive cubic and square numbers (in either order).

The numbers that are not the difference of two squares are 2, 6, 10, 14, 18, ... (Sloane's A016825; Wells 1986, p. 76).

A square number can be the concatenation of two squares, as in the case and giving . The first few numbers that are neither square nor the sum of a square and a prime are 10, 34, 58, 85, 91, 130, 214, ... (Sloane's A020495).

It is conjectured that, other than , and , there are only a finite number of squares having exactly two distinct nonzero digits (Guy 1994, p. 262). The first few such n are 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, ... (Sloane's A016070), corresponding to of 16, 25, 36, 49, 64, 81, 121, ... (Sloane's A018884).

The following table gives the first few numbers which, when squared, give numbers composed of only certain digits. The values of n such that contains exactly two different digits are given by 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, ... (Sloane's A016069), whose squares are 16, 25 36, 49, 64, ... (Sloane's A018885). The only known square number composed only of the digits 7, 8, and 9 is 9. Based on a discussion in rec.puzzles, Vardi (1991) considered numbers composed only of the square digits: 1, 4, and 9. It is conjectured that there are only finitely many, and the largest known is

6480702115891070212
	(32)

found by G. Jacobson and D. Applegate (rec.puzzles FAQ).

Brown numbers are pairs (m, n) of integers satisfying the condition of Brocard's problem, i.e., such that

where is a factorial. Only three such numbers are known: (5,4), (11,5), (71,7). Erdos conjectured that these are the only three such pairs.

Either or has a solution in positive integers iff, for some n, , where is a Fibonacci number and is a Lucas number (Honsberger 1985, pp. 114-118).

The smallest and largest square numbers containing the digits 1 to 9 are

The smallest and largest square numbers containing the digits 0 to 9 are

(Madachy 1979, p. 159). The smallest and largest square numbers containing the digits 1 to 9 twice each are

and the smallest and largest containing 1 to 9 three times are

(Madachy 1979, p. 159).

Madachy (1979, p. 165) also considers number that are equal to the sum of the squares of their two "halves" such as

in addition to a number of others.

Antisquare Number, Biquadratic Number, Brocard's Problem, Brown Numbers, Cannonball Problem, Catalan's Conjecture, Centered Square Number, Clark's Triangle, Cubic Number, Diophantine Equation, Fermat's 4n Plus 1 Theorem, Greedy Algorithm, Gross, Heptagonal Square Number, Lagrange's Four-Square Theorem, Landau-Ramanujan Constant, Octagonal Square Number, Partition, Pentagonal Square Number, Pseudosquare, Pyramidal Number, Squarefree, Square Triangular Number, Sum of Squares Function, Waring's Problem