Square Number

A square number, also called a perfect square, is a figurate number of the form , where is an integer. The square numbers for , 1, ... are 0, 1, 4, 9, 16, 25, 36, 49, ... (OEIS 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 th square number by

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

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

as can be seen in the above diagram, in which the st triangular number is represented by the white triangles, the th 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,

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 require four squares to represent them, the least integer such that every positive integer beyond a certain point can be represented by at most squares is given by .

The number of representations of a number by 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, ... (OEIS A002828), and the numbers of unordered partitions of the numbers 1, 2, 3, ... into positive squares are 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, ... (OEIS A001156). A brute-force algorithm for enumerating the square partitions of uses backtracking through nonincreasing positive square parts. However, this approach rapidly becomes impractical since the number of representations grows extremely rapidly with , as shown in the following table.

For a positive integer , the th nonsquare number is given by

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

The only numbers that are simultaneously positive square and square pyramidal (the cannonball problem) are and , corresponding to and (Ball and Coxeter 1987, p. 59; Ogilvy 1988; Dickson 2005, p. 25), 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 positive numbers that are square and tetrahedral are , , and (giving , , and ), as proved by Meyl (1878; cited in Guy 1994, p. 147; Dickson 2005, p. 25). 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 , where is a nonnegative integer and , 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 , 1, ..., 9 (where numbers with more than 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 , 1, 2, 3, and 4, since

has the same last two digits as (with the one additional possibility that 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 stands for an even number and for an odd number. Additionally, a necessary (but not sufficient) condition for a positive number to be square is that its digital root be 1, 4, 7, or 9. The digital roots of the first few positive squares are 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, ... (OEIS A056992), while the list of numbers having digital roots 1, 4, 7, or 9 is 1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, ... (OEIS A056991).

This property of square numbers was referred to in a "puzzler" feature of a March 2008 broadcast of the NPR radio show "Car Talk." In this Puzzler, a son tells his father that his computer and math teacher assigned the class a problem to determine if a number is a perfect square. Each student is assigned a particular number, and the students are supposed to write a software program to determine the answer. The son's assigned number was . While the father thinks this is a hard problem, a bystander listening to the conversation states that the teacher gave the son an easy number and the bystander can give the answer immediately. The question is what does this third person know? The answer is that the number ends in the digit "2," which is not one of the possible last digits for a square number.

The following table gives the possible residues mod for square numbers for to 20. The quantity gives the number of distinct residues for a given .

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 in (i.e., mod ) can be calculated. Let be an odd prime. Then is the multiplicative function given on prime powers by the following formulas, where the exponent is a positive integer:

is related to the number of unit quadratic residues in by

for (Stangl 1996).

For a perfect square , or 1 for all odd primes where is the Legendre symbol. For a fixed odd prime , the pseudosquare is the least nonsquare positive integer congruent to 1 (mod 8) for which for every odd prime .

Hirschhorn (2001) proved Gosper's conjecture that every sum of four distinct odd squares is the sum of four distinct even squares using the identity

where , , , and are integers. Hirschhorn (2001) also showed that a sum of four distinct even squares is a sum of four distinct odd squares when .

A prime number can be written as the sum of two squares iff is not divisible by 4 (Fermat's 4n+1 theorem). An arbitrary positive integer 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 to the requirement that all the odd prime factors of the squarefree part of 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, ... (OEIS A001481). Letting be the fraction of numbers that are expressible as the sum of two squares,

and

where 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 , 2, 3, and 4 squares to represent them as a sum (Wells 1986, p. 70).

Fermat's 4n+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 (OEIS A001422; Guy 1994; Savin 2000). The following numbers require exactly five distinct positive 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 (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 with exactly unordered representations as a sum of positive squares, with repetition allowed and signs ignored. For example,

can be represented in two ways () by two squares ().

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

Four distinct nonzero integers in arithmetic progression can have a square product. For example, (, , 1, 3) gives (Le Lionnais 1983, p. 53). It is possible to have three squares in nonconstant arithmetic progression, but not four (Dickson 2005, pp. 435-440). A family of three-term progressions , , and is given by

where are coprime integers (Dickson 2005, pp. 437-438). Multiplying , , and by a common positive integer gives further examples. Every nonconstant three-term progression of positive squares, ordered with , can be associated with a Pythagorean triple by

(Robertson 1996).

Catalan's conjecture, proved by Preda Mihăilescu (2004), states that 8 and 9 ( and ) are the only consecutive powers of positive integers greater than 1 with exponents greater than 1, i.e., the only solution to Catalan's Diophantine problem. It is also known that 8 and 9 are the only consecutive cubic and square numbers greater than 1 (in either order).

The numbers that are not the difference of two squares are 2, 6, 10, 14, 18, ... (OEIS 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, ... (OEIS A020495).

It is conjectured that, other than , and , there are only a finite number of squares having exactly two distinct decimal digits (Guy 1994, p. 262). The first few such are 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, ... (OEIS A016070), corresponding to of 16, 25, 36, 49, 64, 81, 121, .... These squares occur in OEIS A018884 after its first three terms 1, 4, and 9. That sequence allows at most two distinct digits and excludes squares ending in 0.

The following table gives the first few numbers which, when squared, give numbers composed of only certain digits. The values of such that contains exactly two different digits are given by 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, ... (OEIS A016069), whose squares are 16, 25, 36, 49, 64, .... The sequence OEIS A018885 allows at most two distinct digits, so it also includes the one-digit squares 0, 1, 4, and 9. In the row for the digits 2, 4, and 8 below, OEIS A027679 requires all three digits to occur and begins with the third displayed root, 168.

For three digits, an extreme example containing only the digits 7, 8, and 9 is

Mishima's list of sporadic solutions, last updated in 2008, gives no square containing exactly the three digits 0, 1, and 3 or 6, 7, and 8, and gives one listed solution for each of the digit sets 019, 039, 056, 079, 568, and 789. These search results do not establish uniqueness among all squares. A listed 52-digit example is

(Mishima 2008).

Brown numbers are pairs of positive 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). Erdős conjectured that these are the only three such pairs.

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

The smallest and largest square numbers containing each of the digits 1 to 9 exactly once are

The smallest and largest square numbers containing each of the digits 0 to 9 exactly once 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 numbers that are equal to the sum of the squares of their two "halves" such as

in addition to a number of others.