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) |

It is usual to consider only ``reduced'' (or ``primitive'') solutions in which and are Relatively Prime, since other solutions can be generated trivially from the primitive ones. For primitive solutions, one of or must be Even, 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) |

(9) |

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

(10) |

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

(11) |

(12) |

(13) |

(14) |

For a Pythagorean triple (, , ),

(15) |

(16) | |||

(17) | |||

(18) |

(Robertson 1996).

The Area of a Triangle corresponding to the Pythagorean triple
is

(19) |

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

(20) |

(21) |

(22) |

(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* Right
Triangle, write its factorization as

(24) |

(25) |

(26) |

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

(27) |

There are 50 Pythagorean triples with Hypotenuse less than 100, the first few of which, sorted by increasing , are , , , , , , , , , , , , , ... (Sloane's A046083, A046084, and A046085). Of these, only 16 are primitive triplets with Hypotenuse less than 100: , , , , , , , , , , , , , , , and (Sloane's A046086, 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 with
Hypotenuse less than be denoted . Then, as the following table suggests and Lehmer (1900) proved, the
number 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 be
consecutive integers is

(29) |

(30) |

(31) | |||

(32) |

then gives the Pell Equation

(33) |

(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, 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

(41) |

(42) |

(43) | |||

(44) |

The first few Leg-Leg triplets are (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ... (Sloane's A046089, A046090, and A046091).

Leg-Hypotenuse twin triples
occur whenever

(45) |

(46) |

(47) |

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 three
sets of generators as

(50) | |||

(51) |

(52) | |||

(53) |

(54) | |||

(55) |

Then the Right Triangle generated by each triple ( ) has common Area

(56) |

One can also find quartets of Right Triangles with the same Area. The Quartet having smallest 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 of
the Legs were Squares. 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 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

(63) |

(64) | |||

(65) |

(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).

**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.

© 1996-9

1999-05-26