It is easy to see that every natural number $n$ can be written in a unique way $n = a+b$ where $gcd(a,b)=1$, $b>a$ and $b-a$ is minimal with this property. For instance if $n$ is odd the representation is given by $n = \frac{n-1}{2}+\frac{n+1}{2}$. Define for every natural number $n$ the following polynomial:
If $n=0(3)$, then $p_n(x) = \frac{n}{3}x+\frac{n}{3}$
If $n=1(3)$, then $p_n(x) = \frac{n-1}{3}x+\frac{n+2}{3}$
If $n=2(3)$, then $p_n(x) = \frac{n+1}{3}x+\frac{n-2}{3}$
(1) Is it true, that if $c\ge3$, $c=a+b$ is the decomposition in coprime integers then $p_c(x) = p_a(x) + p_b(x)$?
(2) From (1) it would follow as in the proof of Snyder of the Mason-Stothers theorem, that: $gcd(a,a')*gcd(b,b')*gcd(c,c') | a'b-b'a = a'c - ac' = bc'-b'c$ where $n' = p_n(x)'$.
(3) Is it true, that: $gcd(a,a')*gcd(b,b')*gcd(c,c') = \pm ( a'b-b'a) $ ?
(4) This might seem a bit far-fetched but I did some computer experiments and it seems that in the situation of (1) we have: $c \le rad(abc)-1$ Does somebody know, if this inequality has a chance to be true or not?
Thanks for your help!
