A twin Pythagorean triple is a Pythagorean triple for which two values are consecutive integers. By definition,
twin triplets are therefore primitive triples. Of the 16 primitive triples with hypotenuse
less than 100, seven are twin triples. The first few twin triples, sorted by increasing
,
are (3, 4, 5), (5, 12, 13), (7, 24, 25), (20, 21, 29), (9, 40, 41), (11, 60, 61),
(13, 84, 85), (15, 112, 113), ....
The numbers of twin triples with hypotenuse less than 10, ,
, ... are 1, 7, 24, 74, ... (OEIS A101903).
The first few leg-leg triplets are (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ... (OEIS A001652,
A046090, and A001653).
A closed form is available for the th such pair. Consider the general reduced solution
, then the requirement that the legs
be consecutive integers is
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(1)
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Rearranging gives
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(2)
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Defining
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(3)
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(4)
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then gives the Pell equation
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(5)
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Solutions to the Pell equation are given by
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(6)
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(7)
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so the lengths of the legs and
and the hypotenuse
are
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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Denoting the length of the shortest leg by then gives
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(17)
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(18)
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(Beiler 1966, pp. 124-125 and 256-257), which cannot be solved exactly to give
as a function of
.
However, the approximate number of leg-leg twin triplets less than a given value of
can be found by noting that the second term in the denominator of
is a small number to the power
and can therefore be dropped, leaving
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(19)
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(20)
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Solving for
gives
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(21)
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(22)
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(23)
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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), ... (OEIS A005408, A046092,
and A001844). Leg-hypotenuse twin triples occur whenever
|
(24)
|
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(25)
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that is to say when , in which case the hypotenuse
exceeds the even leg by unity
and the twin triplet is given by
. The number of leg-hypotenuse triplets
with hypotenuse
is therefore given by
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(26)
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where
is the floor function. The first few values are
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, ... (OEIS A095861).
The numbers of leg-hypotenuse triples less than 10,
, ... are 1, 6, 21, 70, 223, 706, 101904, ... (OEIS A101904).
The total number of twin triples less than
is therefore approximately given by
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(27)
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(28)
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where one has been subtracted to avoid double counting of the leg-leg-hypotenuse double-twin .