I still do not have the answer but maybe my hints will cause
someone to get a formal proof.
Using excel and sloan I arrived to the following recursion formula:
A(n)= B*A(n-1)-A(n-2)
B , A(0)and A(1) are dependenent on the D in Pell's equation :
D=2 B=6 A(0)=0 A(1)=0 ... EQ2
D=3 B=4 A(0)=1 A(1)=4 ... EQ3
D=6 B=10 A(0)=0 A(1)=2 .. EQ6
And the corresponding series are:
0,2,12,70,408 .....e.g 2*70*70+1 is a square of 99
1,4,15,56,209,780 ....e.g 3*15*15+1 is a square of 26
0,2,20,198,1960 .....e.g 6*20*60+1 is a square of 49
The 1st and the third series have 0 as an offset and consist of even numbers only- the second alternately even and odd.
By visual or computer-aided inspection of N members one can assume that there is no a common x present in all three of them, but that does not form a proof.
If some one still wants to make such a comparison -the
recursive formulas might be helpful. However the numbers grow very fast and the comparison would be useful ONLY if a counter example exist i.e. the assumption is wrong.
Still, the way to prove it is not yet clear to me.
still nlng way to go
