n and m natural numbers
Lorentz-friends if the Lorentz-transformation of the two Egyptian fractions 1/n
and 1/m is also an Egyptian fraction.
As
a reminder: Lorentz-transformation of n and m is (n+m)/(+nm).
So
n and m are Lorentz-friend if and only if exists natural number k that
(1/n+1/m)/(1+1/nm)
= 1/k
Like
the light, 1 is Lorentz-friend with all naturals (the Lorentz transformation of
1 and m is 1)! There is concern about this kind of indiscriminate friendship,
so we would rather call it pseudo-friendship and ignore it simply when talking
about Lorentz friendship.
A
perfect antipode is 2 that has no true Lorentz friends.
Another
extremely interesting phenomenon is that every odd number and the following odd
are Lorentz friends. In fact:
(1/2n-1)+1/(2n+1))/(1+1/(2n-1)(2n+1))
= 1/n
It
follows from the above that every odd number greater than 3 has at least two
Lorentz friends.
Is
it possible to know how many Lorentz friends a natural number has? What
relationships and characteristics can be established?
Every
natural number greater than 2 seems to have at least one, every odd number to has
at least two Lorentz friends, but not always more. However,
there are many examples where a number has 3 or more Lorentz friends.
For
example 16 have 3 friends: 35, 69 and 239. The nearby 18 only have one friend:
305.
There
are bigger achievements. So 31 have 13 friends: 9, 17, 29, 33, 49, 65, 89, 129,
161, 209, 289, 449, 929.
We
can define a sequence a(n) that gives n the smallest Lorentz friend which
largest that n. Clearly, if n is odd a(n)=n+2, it can make the sequence a bit
dull. Let's hope that some interesting things will come. For the sake of order,
it should be agreed that a(1)=1, and if n has no friends, then a(n)=0. So the
sequence looks like this:
1,
0, 5, 11, 7, 29, 9, 13, 11, 23, 13, 131, 15, 25, 17, 35, 19, 305, 21, 37, 23,…
To
be continued.
* * *
