which is not a Pythagorean triplet of the following?
(8, 15, 17)
(11, 60, 61)
(15, 20, 25)
(10, 15, 20)
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A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c. By the Pythagorean theorem, this is equivalent to finding positive integers a, b, and c satisfying
a^2+b^2=c^2.
(1)
The smallest and best-known Pythagorean triple is (a,b,c)=(3,4,5). The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.
PythagoreanTriples
Plots of points in the (a,b)-plane such that (a,b,sqrt(a^2+b^2)) is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of a and b, and are therefore symmetric about both the x- and y-axes.
PythagoreanTriplesAC
Similarly, plots of points in the (a,c)-plane such that (a,sqrt(c^2-a^2),c) is a Pythagorean triple are shown above for successively larger bounds.
PrimitivePythagoreanTriple
It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which a and b 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 a or b must be even, and the other odd (Shanks 1993, p. 141), with c 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 (a_0,b_0,c_0), three new primitive triples are obtained from
(a_1,b_1,c_1) = (a_0,b_0,c_0)U
(2)
(a_2,b_2,c_2) = (a_0,b_0,c_0)A
(3)
(a_3,b_3,c_3) = (a_0,b_0,c_0)D,
(4)
where
U =[ 1 2 2; -2 -1 -2; 2 2 3]
(5)
A =[ 1 2 2; 2 1 2; 2 2 3]
(6)
D =[-1 -2 -2; 2 1 2; 2 2 3].
(7)
Hall (1970) and Roberts (1977) prove that (a,b,c) is a primitive Pythagorean triple iff
(a,b,c)=(3,4,5)M,
(8)
where M is a finite product of the matrices U, A, D. It therefore follows that every primitive Pythagorean triple must be a member of the infinite array
( 7, 24, 25); ( 5, 12, 13) ( 55, 48, 73); ( 45, 28, 53); ( 39, 80, 89); (3, 4, 5) ( 21, 20, 29) ( 119, 120, 169); ( 77, 36, 85); ( 33, 56, 65); ( 15, 8, 17) ( 65, 72, 97); ( 35, 12, 37).
(9)
Pythagoras and the Babylonians gave a formula for generating (not necessarily primitive) triples as
(2m,m^2-1,m^2+1),
(10)
for m>1, which generates a set of distinct triples containing neither all primitive nor all imprimitive triples (and where in the special case m=2, m^2-1<2m).
The early Greeks gave
(v^2-u^2,2uv,u^2+v^2),
(11)
where u and v>u 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 v^2-u^2 and 2uv).
Let F_n be a Fibonacci number. Then
(F_nF_(n+3),2F_(n+1)F_(n+2),F_(n+1)^2+F_(n+2)^2)
(12)
generates distinct Pythagorean triples (Dujella 1995), although not exhaustively for either primitive or imprimitive triples. More generally, starting with positive integers a, b, and constructing the Fibonacci-like sequence {F_n^'} with terms a, b, a+b, a+2b, 2a+3b, ... generates distinct Pythagorean triples
(F_n^'F_(n+3)^',2F_(n+1)^'F_(n+2)^',F_(n+1)^'^2+F_(n+2)^'^2)
(13)
(Horadam 1961), where
F_n^'=1/2[(3a-b)F_n+(b-a)L_n,
(14)
where L_n 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
xy(x^4-y^4)=zw(z^4-w^4)
(15)
(Guy 1994, p. 188).
For a Pythagorean triple (a, b, c),
P_3(a)+P_3(b)=P_3(c),
(16)
where P_3 is the partition function P (Honsberger 1985). Every three-term progression of squares r^2, s^2, t^2 can be associated with a Pythagorean triple (X,Y,Z) by
r = X-Y
(17)
s = Z
(18)
t = X+Y
(19)
(Robertson 1996).
The area of a triangle corresponding to the Pythagorean triple (u^2-v^2,2uv,u^2+v^2) is
A=1/2(u^2-v^2)(2uv)=uv(u^2-v^2).
(20)
Fermat proved that a number of this form can never be a square number.
To find the number L_p(s) of possible primitive triangles which may have a leg (other than the hypotenuse) of length s, factor s into the form
s=p_1^(alpha_1)...p_n^(alpha_n).
(21)
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