Utilisateur:Robert FERREOL

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Elementary properties of primitive Pythagorean triples

General properties

Propriétés d'un triplet Pythagoricien (a, b, c) vérifiant a < b < c :

• un entier exactement parmi a et b est impair et c est impair.
• ${\tfrac {(c-a)(c-b)}{2}}$ est un carré et si le triplet est primitif avec a impair, ${\tfrac {c-a}{2}}$ et $c-b$ sont des carrés.
• réciproque fausse comme le montre (1, 8, 9)
• l'un des nombres a, b, c est un carré.
• l'aire d'un triangle pythagoricien S = ab/2) n'est pas un carré ni le double d'un carré.
• Un entier exactement parmi a et b est multiple de 3.
• Un entier exactement parmi a et b est multiple de 4.
• Un entier exactement parmi a , b et c est multiple de 5.
• abc est multiple de 60 (conséquence des trois propriétés précédentes)
• les facteurs premiers de c sont de la forme.4n + 1, donc c également.
• The area ( is a congruent number divisible by 6.
• In every Pythagorean triangle, 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 r = n(m − n), 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 m² + n², 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 w > x > y > z, respectively. Integers −w, x, y, z satisfy Descartes's Circle Equation. Equivalently, the radius of the outer Soddy circle of any right triangle is equal to its semiperimeter. The outer Soddy center is located at D, where ACBD is a rectangle, ACB the right triangle and AB its hypotenuse.
• Only two sides of a primitive Pythagorean triple can be simultaneously prime because by Euclid's formula for generating a primitive Pythagorean triple, one of the legs must be composite and even. However, only one side can be an integer of perfect power $p\geq 2$ because if two sides were integers of perfect powers with equal exponent $p$ it would contradict the fact that there are no integer solutions to the Diophantine equation $x^{2p}\pm y^{2p}=z^{2}$, with $x$, $y$ and $z$ being pairwise coprime.
• There are no Pythagorean triangles in which the hypotenuse and one leg are the legs of another Pythagorean triangle; this is one of the equivalent forms of Fermat's right triangle theorem.
• Each primitive Pythagorean triangle has a ratio of area, K, to squared semiperimeter, s, that is unique to itself and is given by

${\tfrac {K}{s^{2}}}={\tfrac {n(m-n)}{m(m+n)}}=1-{\tfrac {c}{s}}.$

• No primitive Pythagorean triangle has an integer altitude from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable.
• The set of all primitive Pythagorean triples forms a rooted ternary tree in a natural way; see Tree of primitive Pythagorean triples.
• Neither of the acute angles of a Pythagorean triangle can be a rational number of degrees. (This follows from Niven's theorem.)

Special cases

In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist:

• 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 4k + 2) is part of a primitive Pythagorean triple. (If the integer has the form 4k, one may take n =1 and m = 2k in Euclid's formula; if the integer is 2k + 1, one may take n = k and m = k + 1.)
• 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 longest leg differ by exactly one. Such triples are necessarily primitive and have the form (2n + 1, 2n² + 2n, 2n² + 2n +1). This results from Euclid's formula by remarking that the condition implies that the triple is primitive and must verify (m² + n²) - 2mn = 1. This implies (m – n)² = 1, and thus m = n + 1. The above form of the triples results thus of substituting m for n + 1 in Euclid's formula.
• There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longest leg differ by exactly two. They are all primitive, and are obtained by putting n = 1 in Euclid's formula. More generally, for every integer k > 0, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by 2k². They are obtained by putting n = k in Euclid's formula.
• There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 20² + 21² = 29²; these are generated by Euclid's formula when ${\tfrac {m-n}{n}}$ is a convergent to √2.
• For each natural number k, there exist k Pythagorean triples with different hypotenuses and the same area.
• For each natural number k, there exist at least k different primitive Pythagorean triples with the same leg a, where a is some natural number (the length of the even leg is 2mn, and it suffices to choose a with many factorizations, for example a = 4b, where b is a product of k different odd primes; this produces at least 2^k different primitive triples).
• For each natural number n, there exist at least n different Pythagorean triples with the same hypotenuse.
• There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse c and the sum of the legs a + b. According to Fermat, the smallest such triple has sides 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.
• There exist non-primitive Pythagorean triangles with integer altitude from the hypotenuse. Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles.