Suppose we two equations $$x^2+y^2=r^2$$ and $$(x-a)^2+(y-b)^2=2g^2$$
Where x,y and r are integer variables greater than 0. a,b and g are integer constants greater than 0.
I conjecture that for any given selection of a,b and g, the number of integer solutions of the form $(x,y,r)$ is less than or equal to 3. How would I go about proving or disproving this? Any tips on tackling this problem. Thanks.
How to get circles with more than $3$ such points...
First, consider "generating" circle of the form $$ x_0^2+y_0^2=2g^2, $$ that has many integer points. One of such rather small and "reach on integer points" circles is the circle with $g=1105=5\times 13\times 17$. It has $108$ integer points (for example).
Second, shift this circle at (any random) integer vector $(a,b)$ and check how many its integer points match condition $$x^2+y^2=r^2,$$ where $r$ is integer.
Then for considered circle and for $(a,b)=(10, 1855)$ (for example) one will get system $$ \left\{ \begin{array}{c} (x-10)^2+(y-1855)^2=2\cdot 1105^2,\\ x^2+y^2=r^2, \end{array} \right. $$ which has $4$ such $(x,y,r)$-solutions:
$(x,y,r)=(231,308,385)$,
$(x,y,r)=(377,336,505)$,
$(x,y,r)=(1557,2076,2595)$,
$(x,y,r)=(605,3300,3355)$.
And I believe that there are examples with more than $4$ points.
Ooops... Consider the same $g=1105$, and $(a,b)=(73,1561)$.
This construction gives us $7$ solutions:
$(x,y,r)=(440,42,442)$,
$(x,y,r)=(882,224,910)$,
$(x,y,r)=(1410,752,1598)$,
$(x,y,r)=(1592,1194,1990)$,
$(x,y,r)=(1634,1488,2210)$,
$(x,y,r)=(1224,2618,2890)$,
$(x,y,r)=(1130,2712,2938)$.
Update:
for $g=32045=5\times 13\times 17\times 29$ (circle with $324$ integer points)
one can find $(a,b)=(823,45311)$, which give us $15$ solutions:
$(x,y,r)=(12720, 1582, 12818)$,
$(x,y,r)=(25382, 7224, 26390)$,
$(x,y,r)=(32144, 12558, 34510)$,
$(x,y,r)=(33576, 13990, 36374)$,
$(x,y,r)=(38910, 20752, 44098)$,
$(x,y,r)=(44552, 33414, 55690)$,
$(x,y,r)=(46134, 44488, 64090)$,
$(x,y,r)=(46040, 48342, 66758)$,
$(x,y,r)=(45080, 55062, 71162)$,
$(x,y,r)=(40062, 67984, 78910)$,
$(x,y,r)=(35064, 74998, 82790)$,
$(x,y,r)=(30510, 79552, 85202)$,
$(x,y,r)=(23496, 84550, 87754)$,
$(x,y,r)=(10574, 89568, 90190)$,
$(x,y,r)=( 3854, 90528, 90610)$.