“Square Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Square Number

释义

Square Number

A Figurate Number of the form , where is an Integer. A square number is also called a PerfectSquare. The first few square numbers are 1, 4, 9, 16, 25, 36, 49, ... (Sloane's A000290). The Generating Function givingthe square numbers is

(1)

The th nonsquare number is given by

(2)

where

is the A000037).

The only numbers which 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 isequivalent to solving the Diophantine Equation

(3)

(Guy 1994, p. 147).

The only numbers which 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 (, , 1, ..., 9). Then

(4)

so the last digit of

is the same as the last digit of

. The following table gives the last digit of

for

,1, ..., 9. As can be seen, the last digit can be only 0, 1, 4, 5, 6, or 9.

0	1	2	3	4	5	6	7	8	9
0	1	4	9	_6	_5	_6	_9	_4	_1

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

(5)




			(6)

so the last two digits are given by

. But since the last digit must be 0, 1, 4, 5, 6, or 9, the followingtable exhausts all possible last two digits.

c
	0	1	2	3	4	5	6	7	8	9
1	01	21	41	61	81	_01	_21	_41	_61	_81
4	16	96	_76	_56	_36	_16	_96	_76	_56	_36
5	25	_25	_25	_25	_25	_25	_25	_25	_25	_25
6	36	_56	_76	_96	_16	_36	_56	_76	_96	_16
9	81	_61	_41	_21	_01	_81	_61	_41	_21	_01

The only possibilities are 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, and 96, whichcan be summarized succinctly as 00, , , 25, , and , where stands for an Even Number and for anOdd Number. Additionally, unless the sum of the digits of a number is 1, 4, 7, or 9, it cannot be a square number.

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


2	2	0, 1
3	2	0, 1
4	2	0, 1
5	3	0, 1, 4
6	4	0, 1, 3, 4
7	4	0, 1, 2, 4
8	3	0, 1, 4
9	4	0, 1, 4, 7
10	6	0, 1, 4, 5, 6, 9
11	6	0, 1, 3, 4, 5, 9
12	4	0, 1, 4, 9
13	7	0, 1, 3, 4, 9, 10, 12
14	8	0, 1, 2, 4, 7, 8, 9, 11
15	6	0, 1, 4, 6, 9, 10
16	4	0, 1, 4, 9
17	9	0, 1, 2, 4, 8, 9, 13, 15, 16
18	8	0, 1, 4, 7, 9, 10, 13, 16
19	10	0, 1, 4, 5, 6, 7, 9, 11, 16, 17
20	6	0, 1, 4, 5, 9, 16

In general, the Odd squares are congruent to 1 (mod 8) (Conway and Guy 1996). Stangl (1996) gives an explicit formula bywhich the number of squares in (i.e., mod ) can be calculated. Let be an Odd Prime. Then is the Multiplicative Function given by

			(7)
			(8)
			(9)
			(10)
			(11)

is related to the number

of Quadratic Residues in

(12)

for

(Stangl 1996).

For a perfect square , or 1 for all Odd Primes where is the Legendre Symbol. A number which is not a perfect square but which satisfies this relationship is called a Pseudosquare.

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 permutations of is repeatedapplication of the Greedy Algorithm. However, this approach rapidly becomes impractical since the number of representationsgrows extremely rapidly with , as shown in the following table.

	Square Partitions
10	4
50	104
100	1116
150	6521
200	27482

Every Positive integer is expressible as a Sum of (at most) square numbers (Waring's Problem).(Actually, the basis set is , so 49 need never be used.) Furthermore, aninfinite number of require four squares to represent them, so the related quantity (the least Integer such that every Positive Integer beyond a certain point requires squares) is given by .

Numbers expressible as the sum of two squares are those whose Prime Factors are of the form taken to an Even Power. Numbers expressible as the sum of three squares are those not of the form for . The following table gives the first few numbers which require , 2, 3, and 4 squares to represent them as asum.

	Sloane	Numbers
1	Sloane's A000290	1, 4, 9, 16, 25, 36, 49, 64, 81, ...
2	Sloane's A000415	2, 5, 8, 10, 13, 17, 18, 20, 26, 29, ...
3	Sloane's A000419	3, 6, 11, 12, 14, 19, 21, 22, 24, 27, ...
4	Sloane's A004215	7, 15, 23, 28, 31, 39, 47, 55, 60, 63, ...

The Fermat 4n+1 Theorem guarantees that every Prime of the form is a sum of two SquareNumbers in only one way.

There are only 31 numbers which 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). All numbers can be expressed as the sum of at most five distinct squares, and only

(13)

and

(14)

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

(15)

The following table gives the numbers which can be represented in

different ways as a sum of

squares. For example,

can be represented in two ways (

) by two squares (

		Sloane	Numbers
1	1	Sloane's A000290	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
2	1	Sloane's A025284	2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, ...
2	2	Sloane's A025285	50, 65, 85, 125, 130, 145, 170, 185, 200, ...
3	1	Sloane's A025321	3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, ...
3	2	Sloane's A025322	27, 33, 38, 41, 51, 57, 59, 62, 69, 74, 75, ...
3	3	Sloane's A025323	54, 66, 81, 86, 89, 99, 101, 110, 114, 126, ...
3	4	Sloane's A025324	129, 134, 146, 153, 161, 171, 189, 198, ...
4	1	Sloane's A025357	4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, ...
4	2	Sloane's A025358	31, 34, 36, 37, 39, 43, 45, 47, 49, 50, 54, ...
4	3	Sloane's A025359	28, 42, 55, 60, 66, 67, 73, 75, 78, 85, 95, 99, ...
4	4	Sloane's A025360	52, 58, 63, 70, 76, 84, 87, 91, 93, 97, 98, 103, ...

The number of Integers which are squares or sums of two squares is

(16)

where

(17)

(Landau 1908; Le Lionnais 1983, p. 31). The product of four distinct Nonzero Integers inArithmetic Progression is square only for (

, 1, 3), giving

(Le Lionnais 1983, p. 53). Itis possible to have three squares in Arithmetic Progression, but not four (Dickson 1952, pp. 435-440). If thesenumbers are

, and

, there are Positive Integers

and

such that

			(18)
			(19)
			(20)

where

and one of

, or

is Even (Dickson 1952, pp. 437-438). Every three-term progression of squarescan be associated with a Pythagorean Triple

) by

			(21)
			(22)
			(23)

(Robertson 1996).

Catalan's Conjecture states that 8 and 9 ( and ) are the only consecutive Powers (excluding 0and 1), i.e., the only solution to Catalan's Diophantine Problem. This Conjecture has not yet been proved orrefuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the Conjecture nothold. It is also known that 8 and 9 are the only consecutive Cubic and square numbers (in eitherorder).

A square number can be the concatenation of two squares, as in the case and giving .

It is conjectured that, other than , and , there are only a Finitenumber of squares having exactly two distinct Nonzero Digits (Guy 1994, p. 262). The firstfew such 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 only knownsquare number composed only of the digits 7, 8, and 9 is 9. Vardi (1991) considers numbers composed only of the square digits:1, 4, and 9.

Digits	Sloane	,
1, 2, 3	Sloane's A030175	1, 11, 111, 36361, 363639, ...Sloane's A
	Sloane's A030174	1, 121, 12321, 1322122321, ...Sloane's A
1, 4, 6	Sloane's A027677	1, 2, 4, 8, 12, 21, 38, 108, ...Sloane's A
	Sloane's A027676	1, 4, 16, 64, 144, 441, 1444, ...Sloane's A
1, 4, 9	Sloane's A027675	1, 2, 3, 7, 12, 21, 38, 107, ...Sloane's A
	Sloane's A006716	1, 4, 9, 49, 144, 441, 1444, 11449, ...Sloane's A
2, 4, 8	Sloane's A027679	2, 22, 168, 478, 2878, 210912978, ...Sloane's A
	Sloane's A027678	4, 484, 28224, 228484, 8282884, ...Sloane's A
4, 5, 6	Sloane's A030177	2, 8, 216, 238, 258, 738, 6742, ...Sloane's A
	Sloane's A030176	4, 64, 46656, 56644, 66564, ...

Brown Numbers are pairs of Integers satisfying the condition of Brocard's Problem,i.e., such that

(24)

where

is a Factorial. Only three such numbers are known: (5,4), (11,5), (71,7). Erdös conjectured that theseare 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 the digits 1 to 9 are

(25)

(26)

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

(27)

(28)

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

(29)

(30)

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

(31)

(Madachy 1979, p. 159).

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

			(32)
			(33)
			(34)
			(35)

in addition to a number of others.

See also Antisquare Number, Biquadratic Number, Brocard's Problem, Brown Numbers, CannonballProblem, Catalan's Conjecture, Centered Square Number, Clark's Triangle, Cubic Number,Diophantine Equation, Fermat 4n+1 Theorem, Greedy Algorithm, Gross, Lagrange's Four-SquareTheorem, Landau-Ramanujan Constant, Pseudosquare, Pyramidal Number, r(n),Squarefree, Square Triangular Number, Waring's Problem

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

Bohman, J.; Fröberg, C.-E.; and Riesel, H. ``Partitions in Squares.'' BIT 19, 297-301, 1979.

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

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

Grosswald, E. Representations of Integers as Sums of Squares. New York: Springer-Verlag, 1985.

Guy, R. K. ``Sums of Squares'' and ``Squares with Just Two Different Decimal Digits.'' §C20 and F24 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138 and 262, 1994.

Honsberger, R. ``A Second Look at the Fibonacci and Lucas Numbers.'' Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Lucas, É. Question 1180. Nouv. Ann. Math. Ser. 2 14, 336, 1875.

Lucas, É. Solution de Question 1180. Nouv. Ann. Math. Ser. 2 15, 429-432, 1876.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 159 and 165, 1979.

Meyl, A.-J.-J. Solution de Question 1194. Nouv. Ann. Math. 17, 464-467, 1878.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 77 and 152, 1988.

Pappas, T. ``Triangular, Square & Pentagonal Numbers.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989.

Pietenpol, J. L. ``Square Triangular Numbers.'' Amer. Math. Monthly 69, 168-169, 1962.

Robertson, J. P. ``Magic Squares of Squares.'' Math. Mag. 69, 289-293, 1996.

Stangl, W. D. ``Counting Squares in .'' Math. Mag. 69, 285-289, 1996.

Taussky-Todd, O. ``Sums of Squares.'' Amer. Math. Monthly 77, 805-830, 1970.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 20 and 234-237, 1991.

Watson, G. N. ``The Problem of the Square Pyramid.'' Messenger. Math. 48, 1-22, 1918.

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