Uniqueness in a Diophantine system

Question

THIS TURNS OUT TO BE FALSE. I POSTED AN ANSWER WITH SOME EXAMPLES.

Suppose we have integers $1 < A < B$ with integer $U>0$ and $$ AB = U^2 + 1. $$ Then $A < U < B.$

If we take $$ C = A - 2 U + B, $$ then $$ AC = (U-A)^2 + 1 $$ and $$ BC = (B-U)^2 + 1. $$

The question is about uniqueness: if we have some integer $F > 1$ such that $$ AF = V^2 + 1, $$ $$ BF = W^2 + 1, $$ is it true that $$ F = A - 2 U + B = C? $$

Oh, why Gaussian integers? The conditions say that $A,B,F$ is each the sum of two squares, and the system is something like the Gaussian integer system

$$u \bar{v} = n + i, \; \; \;v \bar{w} = m + i, \; \; \; w \bar{u} = k + i,$$

A positive answer on uniqueness would finish Find integers $(w, x, y, z)$ such that the product of each two of them minus 1 is square.

Usually I would run some long computer experiments to find possible counterexamples and avoid looking foolish, but I have been running something else for two days and wish to continue that one.

Answer 1

apparently there is no uniqueness, and the original problem to which I refer is an open problem...

5  2  13
5  2  925
10  5  29
13  2  5
13  2  25
13  5  2
13  5  34
17  10  53
25  2  13
25  2  41
25  13  2
25  13  74
26  17  85
29  5  10
29  5  58
29  10  5
29  10  73
34  5  13
34  5  65
34  13  5
34  13  89
37  26  125
41  2  25
41  2  61
41  25  2
41  25  130
50  37  173