rational points on one dimensional sphere
be a one dimensional sphere.
We will denote by
Theorem. Let . Then if and only if there exists a Pythagorean triplet (i.e. is a Pythagorean triplet) such that are of the form and .
Proof. ,,” If (for example) and for a Pythagorean triplet , then we have
and thus .
,,” Assume that . Then for some . It follows, that
and this is if and only if
(up to a sign of course). Therefore if and only if is an integer. In this case we have
so is a Pythagorean triplet, which completes the proof.
Corollary. The rational sphere is an infinite set.
Proof. Let be natural numbers such that is fixed and even. Let run through primes. Then (due to the theorem in parent entry)
is a Pythagorean triplet. Let
Ir follows from the theorem, that there exists such that . It is easy to see, that if and only if and thus we generated infinitely many rational points on sphere. This completes the proof.
|Title||rational points on one dimensional sphere|
|Date of creation||2013-03-22 19:07:49|
|Last modified on||2013-03-22 19:07:49|
|Last modified by||joking (16130)|