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.