# prime factors of Pythagorean hypotenuses

The possible hypotenuses^{} of the
Pythagorean triangles^{} (http://planetmath.org/PythagoreanTriangle) form the
infinite^{} sequence^{}

$$5,\mathrm{\hspace{0.17em}10},\mathrm{\hspace{0.17em}13},\mathrm{\hspace{0.17em}15},\mathrm{\hspace{0.17em}17},\mathrm{\hspace{0.17em}20},\mathrm{\hspace{0.17em}25},\mathrm{\hspace{0.17em}26},\mathrm{\hspace{0.17em}29},\mathrm{\hspace{0.17em}30},\mathrm{\hspace{0.17em}34},\mathrm{\hspace{0.17em}35},\mathrm{\hspace{0.17em}37},\mathrm{\hspace{0.17em}39},\mathrm{\hspace{0.17em}40},\mathrm{\hspace{0.17em}41},\mathrm{\hspace{0.17em}45},\mathrm{\dots}$$ |

the mark of which is
http://oeis.org/search?q=a009003&language^{}=english&go=SearchA009003 in the corpus of the integer
sequences of http://oeis.org/OEIS. This sequence has the subsequence A002144

$$5,\mathrm{\hspace{0.17em}13},\mathrm{\hspace{0.17em}17},\mathrm{\hspace{0.17em}29},\mathrm{\hspace{0.17em}37},\mathrm{\hspace{0.17em}41},\mathrm{\hspace{0.17em}53},\mathrm{\hspace{0.17em}61},\mathrm{\hspace{0.17em}73},\mathrm{\hspace{0.17em}89},\mathrm{\hspace{0.17em}97},\mathrm{\hspace{0.17em}101},\mathrm{\hspace{0.17em}109},\mathrm{\hspace{0.17em}113},\mathrm{\hspace{0.17em}137},\mathrm{\dots}$$ |

of the odd Pythagorean primes.

Generally, the hypotenuse $c$ of a Pythagorean triangle
(Pythagorean triple^{}) may be characterised by being the
contraharmonic mean

$$c=\frac{{u}^{2}+{v}^{2}}{u+v}$$ |

of some two different integers $u$ and $v$ (as has been shown in the parent entry), but also by the

Theorem. A positive integer $c$ is the length of the
hypotenuse of a Pythagorean triangle if and only if at least
one of the prime factors^{} of $c$ is of the form $4n+1$.

Lemma 1. All prime factors of the hypotenuse $c$ in a primitive Pythagorean triple are of the form $4n+1$.

This can be proved here by making the antithesis that there exists a prime $4n-1$ dividing $c$. Then also

$$4n-1\mid {c}^{2}={a}^{2}+{b}^{2}=(a+ib)(a-ib)$$ |

where $a$ and $b$ are the catheti in the triple. But $4n-1$
is prime also in the ring $\mathbb{Z}[i]$ of the Gaussian
integers, whence it must divide at least one of the factors
$a+ib$ and $a-ib$. Apparently, that would imply that
$4n-1$ divides both $a$ and $b$. This means that the
triple $(a,b,c)$ were not primitive, whence
the antithesis is wrong and the lemma true. $\mathrm{\square}$

Also the converse^{} is true in the following form:

Lemma 2. If all prime factors of a positive integer $c$ are of the form $4n+1$, then $c$ is the hypotenuse in a Pythagorean triple. (Especially, any prime $4n+1$ is found as the hypotenuse in a primitive Pythagorean triple.)

Proof. For proving this, one can start from Fermat’s
theorem, by which the prime numbers^{} of such form are sums of
two squares (see the
http://en.wikipedia.org/wiki/Proofs_of_Fermat's_theorem_on_sums_of_two_squaresTheorem on sums of two squares by Fermat).
Since the sums of two squares form a set closed under
multiplication^{}, now also the product $c$ is a sum of two
squares, and similarly is ${c}^{2}$, i.e. $c$ is the hypotenuse
in a Pythagorean triple. $\mathrm{\square}$

Proof of the Theorem. Suppose that $c$ is the hypotenuse of a Pythagorean triple $(a,b,c)$; dividing the triple members by their greatest common factor we get a primitive triple $({a}^{\prime},{b}^{\prime},{c}^{\prime})$ where ${c}^{\prime}\mid c$. By Lemma 1, the prime factors of ${c}^{\prime}$, being also prime factors of $c$, are of the form $4n+1$.

On the contrary, let’s suppose that a prime factor $p$ of
$c=pd$ is of the form $4n+1$. Then Lemma 2 guarantees
a Pythagorean triple $(r,s,p)$, whence also $(rd,sd,c)$ is
Pythagorean and $c$ thus a hypotenuse. $\mathrm{\square}$

Title | prime factors of Pythagorean hypotenuses |
---|---|

Canonical name | PrimeFactorsOfPythagoreanHypotenuses |

Date of creation | 2014-01-31 10:48:20 |

Last modified on | 2014-01-31 10:48:20 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11D09 |

Classification | msc 51M05 |

Classification | msc 11E25 |