“Pythagorean Triple”的概念、定义、习题解题方法及翻译-数学辞典

单词

Pythagorean Triple

释义

Pythagorean Triple

A Pythagorean triple is a Triple of Positive Integers , , and such that a RightTriangle exists with legs and Hypotenuse . By the Pythagorean Theorem, this is equivalent to findingPositive Integers , , and satisfying

(1)

The smallest and best-known Pythagorean triple is

It is usual to consider only ``reduced'' (or ``primitive'') solutions in which and are Relatively Prime, sinceother solutions can be generated trivially from the primitive ones. For primitive solutions, one of or must beEven, and the other Odd (Shanks 1993, p. 141), with always Odd. In addition, in every Pythagorean triple,one side is always Divisible by 3 and one by 5.

Given a primitive triple , three new primitive triples are obtained from

			(2)
			(3)
			(4)

where

			(5)
			(6)
			(7)

Roberts (1977) proves that

is a primitive Pythagorean triple Iff

(8)

where

is a Finite Product of the Matrices

. Ittherefore follows that every primitive Pythagorean triple must be a member of the Infinite array

(9)

For any Pythagorean triple, the Product of the two nonhypotenuse Legs (i.e., the two smallernumbers) is always Divisible by 12, and the Product of all three sides is Divisible by 60. It is notknown if there are two distinct triples having the same Product. The existence of two such triples corresponds to aNonzero solution to the Diophantine Equation

(10)

(Guy 1994, p. 188).

Pythagoras and the Babylonians gave a formula for generating (not necessarily primitive) triples:

(11)

and Plato

gave

(12)

A general reduced solution (known to the early Greeks) is

(13)

where

and

are Relatively Prime (Shanks 1993, p. 141). Let

be a FibonacciNumber. Then

(14)

is also a Pythagorean triple.

For a Pythagorean triple (, , ),

(15)

where

is the Partition Function P (Garfunkel 1981, Honsberger 1985). Everythree-term progression of Squares

can be associated with a Pythagorean triple

) by

			(16)
			(17)
			(18)

(Robertson 1996).

The Area of a Triangle corresponding to the Pythagorean triple is

(19)

Fermat

proved that a number of this form can never be a Square Number.

To find the number of possible primitive Triangles which may have a Leg (other than theHypotenuse) of length , factor into the form

(20)

The number of such Triangles is then

(21)

i.e., 0 for Singly Even

and 2 to the power one less than the number of distinct prime factorsof

otherwise (Beiler 1966, pp. 115-116). The first few numbers for

, 2, ..., are 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1,2, 1, 0, 2, ... (Sloane's A024361). To find the number of ways

in which a number

can be the Leg (other thanthe Hypotenuse) of a primitive or nonprimitive Right Triangle, write the factorization of

(22)

Then

(23)

(Beiler 1966, p. 116). The first few numbers for , 2, ... are 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, ...(Sloane's A046079).

To find the number of ways in which a number can be the Hypotenuse of a primitive RightTriangle, write its factorization as

(24)

where the

s are of the form

and the

s are of the form

. The number of possible primitive RightTriangles is then

(25)

The first few Primes of the form

are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, ...(Sloane's A002144), so the smallest side lengths which are the hypotenuses of 1, 2, 4, 8, 16, ... primitive right triangles are5, 65, 1105, 32045, 1185665, 48612265, ... (Sloane's A006278). The number of possible primitive or nonprimitiveRight Triangles having

as a Hypotenuse is

(26)

(Beiler 1966, p. 117). The first few numbers for

, 2, ... are 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0,... (Sloane's A046080).

Therefore, the total number of ways in which may be either a Leg or Hypotenuse of a Right Triangle isgiven by

(27)

The values for

, 2, ... are 0, 0, 1, 1, 2, 1, 1, 2, 2, 2, 1, 4, 2, 1, 5, 3, ... (Sloane's A046081). The smallestnumbers

which may be the sides of

general Right Triangles for

, 2, ... are 3, 5, 16,12, 15, 125, 24, 40, ... (Sloane's A006593; Beiler 1966, p. 114).

There are 50 Pythagorean triples with Hypotenuse less than 100, the first few of which, sorted by increasing , are, , , , , , , , ,, , , , ... (Sloane'sA046083,A046084,and A046085).Of these, only 16 are primitive triplets with Hypotenuse less than 100: , , , , , , , , , , , , , , , and (Sloane'sA046086,A046087,and A046088).Of these 16 primitive triplets, seven are twin triplets (defined as triplets for which two members are consecutive integers). The first few twin triplets, sorted by increasing , are, , , , , , , ,....

Let the number of triples with Hypotenuse less than be denoted , and the number of twin triplets withHypotenuse less than be denoted . Then, as the following table suggests and Lehmer (1900) proved, thenumber of primitive solutions with Hypotenuse less than satisfies

(28)


100	16	0.1600	7
500	80	0.1600	17
1000	158	0.1580	24
2000	319	0.1595	34
3000	477	0.1590	41
4000	639	0.1598	47
5000	792	0.1584	52
10000	1593	0.1593	74

Considering twin triplets in which the Legs are consecutive, a closed form is available for the th such pair.Consider the general reduced solution , then the requirement that the Legs beconsecutive integers is

(29)

Rearranging gives

(30)

Defining

			(31)
			(32)

then gives the Pell Equation

(33)

Solutions to the Pell Equation are given by

			(34)
			(35)

so the lengths of the legs

and

and the Hypotenuse

are


			(36)

			(37)

			(38)

Denoting the length of the shortest Leg by

then gives

			(39)
			(40)

(Beiler 1966, pp. 124-125 and 256-257), which cannot be solved exactly to give

as a function of

. However, theapproximate number of leg-leg twin triplets

less than a given value of

can be found by noting that thesecond term in the Denominator of

is a small number to the power

and can therefore be dropped, leaving

(41)

(42)

Solving for

gives


			(43)
			(44)

The first few Leg-Leg triplets are (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ... (Sloane'sA046089,A046090,and A046091).

Leg-Hypotenuse twin triples occur whenever

(45)

(46)

that is to say when

, in which case the Hypotenuse exceeds the Even Leg by unity and the twin tripletis given by

. The number of leg-hypotenuse triplets with hypotenuse less than

is therefore given by

(47)

where

is the Floor Function. The first few Leg-Hypotenuse triples are (3, 4, 5), (5, 12, 13),(7, 24, 25), (9, 40, 41), (11, 60, 61), (13, 84, 85), ... (Sloane'sA005408,A046092,and A046093).

The total number of twin triples less than is therefore approximately given by

			(48)
			(49)

where one has been subtracted to avoid double counting of the leg-leg-hypotenuse double-twin (3,4,5).

There is a general method for obtaining triplets of Pythagorean triangles with equal Areas. Take the threesets of generators as

			(50)
			(51)

			(52)
			(53)

			(54)
			(55)

Then the Right Triangle generated by each triple (

) has common Area

(56)

(Beiler 1966, pp. 126-127). The only Extremum of this function occurs at

. Since

for

,the smallest Area shared by three nonprimitive Right Triangles is given by

,which results in an area of 840 and corresponds to the triplets (24, 70, 74), (40, 42, 58), and (15, 112, 113) (Beiler 1966,p. 126). The smallest Area shared by three primitive Right Triangles is 13123110,corresponding to the triples (4485, 5852, 7373), (1380, 19019, 19069), and (3059, 8580, 9109) (Beiler 1966, p. 127).

One can also find quartets of Right Triangles with the same Area. The Quartet havingsmallest known area is (111, 6160, 6161), (231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458), with Area 341,880(Beiler 1966, p. 127). Guy (1994) gives additional information.

It is also possible to find sets of three and four Pythagorean triplets having the same Perimeter (Beiler 1966,pp. 131-132). Lehmer (1900) showed that the number of primitive triples with Perimeter less than is

(57)

In 1643, Fermat challenged Mersenne to find a Pythagorean triplet whose Hypotenuse and Sum ofthe Fermat found the smallest such solution:

			(58)
			(59)
			(60)

with

			(61)
			(62)

A related problem is to determine if a specified Integer can be the Area of a Right Trianglewith rational sides. 1, 2, 3, and 4 are not the Areas of any Rational-sidedRight Triangles, but 5 is (3/2, 20/3, 41/6), as is 6 (3, 4, 5). The solution to the probleminvolves the Elliptic Curve

(63)

A solution (

) exists if (63) has a Rational solution, in which case

			(64)
			(65)

(Koblitz 1993). There is no known general method for determining if there is a solution for arbitrary

, but a techniquedevised by J. Tunnell in 1983 allows certain values to be ruled out (Cipra 1996).

See also Heronian Triangle, Pythagorean Quadruple, Right Triangle

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 57-59, 1987.

Beiler, A H. ``The Eternal Triangle.'' Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.

Cipra, B. ``A Proof to Please Pythagoras.'' Science 271, 1669, 1996.

Courant, R. and Robbins, H. ``Pythagorean Numbers and Fermat's Last Theorem.'' §2.3 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 40-42, 1996.

Dickson, L. E. ``Rational Right Triangles.'' Ch. 4 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 165-190, 1952.

Dixon, R. Mathographics. New York: Dover, p. 94, 1991.

Garfunkel, J. Pi Mu Epsilon J., p. 31, 1981.

Guy, R. K. ``Triangles with Integer Sides, Medians, and Area.'' §D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.

Hindin, H. ``Stars, Hexes, Triangular Numbers, and Pythagorean Triples.'' J. Recr. Math. 16, 191-193, 1983-1984.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 47, 1985.

Koblitz, N. Introduction to Elliptic Curves and Modular Forms, 2nd ed. New York: Springer-Verlag, pp. 1-50, 1993.

Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 95-104, 1942.

Kramer, K. and Tunnell, J. ``Elliptic Curves and Local Epsilon Factors.'' Comp. Math. 46, 307-352, 1982.

Lehmer, D. N. ``Asymptotic Evaluation of Certain Totient Sums.'' Amer. J. Math. 22, 294-335, 1900.

Roberts, J. Elementary Number Theory: A Problem Oriented Approach. Cambridge, MA: MIT Press, 1977.

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

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 121 and 141, 1993.

Sloane, N. J. A. SequencesA002144/M3823, A005408/M2400, A006278, A006593/M2499, A020882, A024361, A046079, A046080, A046081, A046083, A046084, A046085, A046086, A046087, A046089, A046090, A046091, A046092, and A046093 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html..

Taussky-Todd, O. ``The Many Aspects of the Pythagorean Triangles.'' Linear Algebra and Appl. 43, 285-295, 1982.

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