Pythagorean Triple - Elementary Properties of Primitive Pythagorean Triples

Elementary Properties of Primitive Pythagorean Triples

The properties of a primitive Pythagorean triple (PPT) (a,b,c) with a < b < c (without specifying which of a or b is even and which is odd) include:

• (c − a)(c − b)/2 is always a perfect square. This is particularly useful in checking if a given triple of numbers is a Pythagorean triple, but it is only a necessary condition, not a sufficient one. The triple {6, 12, 18} passes the test that (c − a)(c − b)/2 is a perfect square, but it is not a Pythagorean triple. When a triple of numbers a, b and c forms a primitive Pythagorean triple, then (c minus the even leg) and one-half of (c minus the odd leg) are both perfect squares; however this is not a sufficient condition, as the triple {1, 8, 9} is a counterexample since 12 + 82 ≠ 92.
• The sum of a,b,c is always an even number. In other words,
• At most one of a, b, c is a square. (See Infinite descent#Non-solvability of r2 + s4 = t4 for a proof.)
• There exist infinitely many primitive Pythagorean triples whose hypotenuses are squares of natural numbers.
• There exist infinitely many primitive Pythagorean triples in which one of the legs is the square of a natural number.
• The sum of the hypotenuse and the even leg of a primitive Pythagorean triple is the square of an odd number, and the arithmetic mean of the hypotenuse and the odd leg is a perfect square.
• The area (K = ab/2) is an even congruent number.
• The area of a Pythagorean triangle cannot be the square or twice the square of a natural number.
• Exactly one of a, b is odd; c is odd.
• Exactly one of a, b is divisible by 3.
• Exactly one of a, b is divisible by 4.
• Exactly one of a, b, c is divisible by 5.
• The largest number that always divides abc is 60.
• Exactly one of a, b, (a + b), (b − a) is divisible by 7.
• Exactly one of (a + c), (b + c), (c − a), (c − b) is divisible by 8.
• Exactly one of (a + c), (b + c), (c − a), (c − b) is divisible by 9.
• Exactly one of a, b, (2a + b), |2a − b|, (2b + a), (2b − a) is divisible by 11.
• Exactly one of a, b, c, (2c + a), (2c + b), (2c − a), (2c − b) is divisible by 13.
• All prime factors of c are primes of the form 4n + 1.
• Every integer greater than 2 that is not congruent to 2 mod 4 (in other words, every integer greater than 2 which is not of the form 4n + 2) is part of a primitive Pythagorean triple.
• Every integer greater than 2 is part of a primitive or non-primitive Pythagorean triple, for example, the integers 6, 10, 14, and 18 are not part of primitive triples, but are part of the non-primitive triples 6, 8, 10; 14, 48, 50 and 18, 80, 82.
• There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly one (such triples are necessarily primitive). Generalization: For every odd integer j, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the even leg differ by j 2.
• There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly two. Generalization: For every integer k > 0, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by .
• There exist infinitely many Pythagorean triples in which the two legs differ by exactly one—e.g., .
• If j and k are odd positive integers, not necessarily unequal, there is exactly one primitive Pythagorean triple with
• The hypotenuse of every primitive Pythagorean triangle exceeds the even leg by the square of an odd integer j, and exceeds the odd leg by twice the square of an integer k > 0, from which it follows that:

• There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater than 2.

• For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
• For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number
• For each natural number n, there exist at least n different Pythagorean triples with the same hypotenuse.
• In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers. Specifically, for a primitive triple the radius of the incircle is, and the radii of the excircles opposite the sides m2–n2, 2mn, and the hypotenuse m2+n2 are respectively m(m − n), n(m + n), and m(m + n).
• As for any right triangle, the converse of Thales' theorem says that the diameter of the circumcircle equals the hypotenuse; hence for primitive triples the circumdiameter is, and the circumradius is half of this and thus is rational but non-integer (since m and n have opposite parity).
• When the area of a Pythagorean triangle is multiplied by the curvatures of its incircle and 3 excircles, the result is four positive integers . Integers satisfy Descartes’s Circle Equation.
• There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple.
• The set of all primitive Pythagorean triples forms a rooted ternary tree in a natural way; see Tree of primitive Pythagorean triples.
• There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse c and the sum of the legs a+b. The smallest such triple has a = 4,565,486,027,761; b = 1,061,652,293,520; and c = 4,687,298,610,289. Here a+b = 2,372,1592 and c = 2,165,0172. This is generated by Euclid's formula with parameter values m = 2,150,905 and n = 246,792.
• Each primitive Pythagorean triangle has a ratio of area to squared semiperimeter that is unique to itself and is given by