Pythagorean Triple

A Pythagorean triple is a triple of positive integers , , and  such that a right triangle exists with legs  and hypotenuse . By the Pythagorean theorem, this is equivalent to finding positive integers , , and  satisfying

(1)

The smallest and best-known Pythagorean triple is . The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.

Plots of points in the -plane such that  is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of  and , and are therefore symmetric about both the x- and y-axes.

Similarly, plots of points in the -plane such that  is a Pythagorean triple are shown above for successively larger bounds.

It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which  and  are relatively prime, since other solutions can be generated trivially from the primitive ones. The primitive triples are illustrated above, and it can be seen immediately that the radial lines corresponding to imprimitive triples in the original plot are absent in this figure. For primitive solutions, one of  or  must be even, and the other odd (Shanks 1993, p. 141), with  always odd.

In addition, one side of every Pythagorean triple is divisible by 3, another by 4, and another by 5. One side may have two of these divisors, as in (8, 15, 17), (7, 24, 25), and (20, 21, 29), or even all three, as in (11, 60, 61).

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

			(2)
			(3)
			(4)

where

		(5)
		(6)
		(7)

Hall (1970) and Roberts (1977) prove that  is a primitive Pythagorean triple iff

(8)

where  is a finite product of the matrices , , . It therefore follows that every primitive Pythagorean triple must be a member of the infinite array

(9)

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

(10)

for , which generates a set of distinct triples containing neither all primitive nor all imprimitive triples (and where in the special case , ).

The early Greeks gave

(11)

where  and  are relatively prime and of opposite parity (Shanks 1993, p. 141), which generates a set of distinct triples containing precisely the primitive triples (after appropriately sorting  and ).

Let  be a Fibonacci number. Then

(12)

generates distinct Pythagorean triples (Dujella 1995), although not exhaustively for either primitive or imprimitive triples. More generally, starting with positive integers , , and constructing the Fibonacci-like sequence  with terms , , , , , ... generates distinct Pythagorean triples

(13)

(Horadam 1961), where

(14)

where  is a Lucas number.

For any Pythagorean triple, the product of the two nonhypotenuse legs (i.e., the two smaller numbers) is always divisible by 12, and the product of all three sides is divisible by 60. It is not known if there are two distinct triples having the same product. The existence of two such triples corresponds to a nonzero solution to the Diophantine equation

(15)

(Guy 1994, p. 188).

For a Pythagorean triple (, , ),

(16)

where  is the partition function P (Honsberger 1985). Every three-term progression of squares , ,  can be associated with a Pythagorean triple ) by

			(17)
			(18)
			(19)

(Robertson 1996).

The area of a triangle corresponding to the Pythagorean triple  is

(20)

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 the hypotenuse) of length , factor  into the form

(21)

The number of such triangles is then

(22)

i.e., 0 for singly even  and 2 to the power one less than the number of distinct prime factors of  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, ... (OEIS A024361). To find the number of ways  in which a number  can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle, write the factorization of  as

(23)

Then

(24)

(Beiler 1966, p. 116). Note that  iff  is prime or twice a prime. The first few numbers for , 2, ... are 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, ... (OEIS A046079).

To find the number of ways  in which a number  can be the hypotenuse of a primitive right triangle, write its factorization as

(25)

where the s are of the form  and the s are of the form . The number of possible primitive right triangles is then

(26)

For example,  since

			(27)
			(28)

The values of  for , 2, ... are 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, ... (OEIS A024362). The first few primes of the form  are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, ... (OEIS A002144), so the smallest side lengths which are the hypotenuses of 1, 2, 4, 8, 16, ... primitive right triangles are 5, 65, 1105, 32045, 1185665, 48612265, ... (OEIS A006278).

The number of possible primitive or nonprimitive right triangles having  as a hypotenuse is

			(29)
			(30)

(correcting the typo of Beiler 1966, p. 117, which states that this formula gives the number of non-primitive solutions only), where  is the sum of squares function. For example, there are four distinct integer triangles with hypotenuse 65, since

(31)

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, ... (OEIS A046080). The smallest hypotenuses having  distinct triples are 1, 5, 25, 125, 65, 3125, ... (OEIS A006339). The following table gives the hypotenuses for which there exist exactly  distinct right integer triangles for , 1, ..., 5.

	OEIS	hypotenuses for which there exist  distinct integer triangles
0	A004144	1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, ...
1	A084645	5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, ...
2	A084646	25, 50, 75, 100, 150, 169, 175, 200, 225, ...
3	A084647	125, 250, 375, 500, 750, 875, 1000, 1125, 1375, ...
4	A084648	65, 85, 130, 145, 170, 185, 195, 205, 221, 255, ...
5	A084649	3125, 6250, 9375, 12500, 18750, 21875, 25000, ...

Therefore, the total number of ways in which  may be either a leg or hypotenuse of a right triangle is given by

(32)

The values for , 2, ... are 0, 0, 1, 1, 2, 1, 1, 2, 2, 2, 1, 4, 2, 1, 5, 3, ... (OEIS A046081). The smallest numbers  which may be the sides of  general right triangles for , 2, ... are 3, 5, 16, 12, 15, 125, 24, 40, ... (OEIS A006593; Beiler 1966, p. 114).

There are 50 Pythagorean triples with hypotenuse less than 100, the first few of which, sorted by increasing , are (3, 4, 5), (6, 8,10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34), (21, 28, 35), ... (OEIS A046083, A046084, and A009000).

Of these, only 16 are primitive triplets with hypotenuse less than 100: (3, 4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65), (48, 55, 73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97) (OEIS A046086, A046087, and A020882).

Let the number of triples with hypotenuse  be denoted , the number of triples with hypotenuse  be denoted , and the number of primitive triples less than  be denoted . Then the following table summarizes the values for powers of 10.

	OEIS	, , ...
	A101929	1, 50, 878, 12467, ...
	A101930	2, 52, 881, 12471, ...
	A101931	1, 16, 158, 1593, ...

Lehmer (1900) proved that the number of primitive solutions with hypotenuse less than  satisfies

(33)

(OEIS A086201).

The inradii of the first few primitive Pythagorean triangles ordered by increasing  are given by 1, 2, 3, 3, 6, 5, 4, 10, 5, ... (OEIS A014498).

There is a general method for obtaining triplets of Pythagorean triangles with equal areas. Take the three sets of generators as

			(34)
			(35)
			(36)
			(37)
			(38)
			(39)

Then the right triangle generated by each triple () has common area

(40)

(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).

Right triangles whose areas consist of a single digit include  (area of 6) and  (area of 666666; Wells 1986, p. 89).

In 1643, Fermat challenged Mersenne to find a Pythagorean triplet whose hypotenuse and sum of the legs were squares. Fermat found the smallest such solution:

			(41)
			(42)
			(43)

with

			(44)
			(45)

A related problem is to determine if a specified integer  can be the area of a right triangle with rational sides. 1, 2, 3, and 4 are not the areas of any rational-sided right triangles, but 5 is (3/2, 20/3, 41/6), as is 6 (3, 4, 5). The solution to the problem involves the elliptic curve

(46)

A solution (, , ) exists if (46) has a rational solution, in which case

			(47)
			(48)

(Koblitz 1993). There is no known general method for determining if there is a solution for arbitrary , but a technique devised by J. Tunnell in 1983 allows certain values to be ruled out (Cipra 1996).

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