# Finding coprime integers near a lattice point

I have a list $L$ of ordered pairs $(n,m)$ where $n$ and $m$ are integers. I would like to know which elements $(n,m)$ in $L$ satisfy the property that $\gcd(n+i,m+j) \neq 1$ for $i =-1,0,1$ and $j =-1,0,1.$ For example the point $(55,21)$ has this property since $[(55+i,21+j) ] = [(54,20),(54,21),(54,22),(55,20),(55,21),(55,22),(56,20),(56,21),(56,22)].$ I have tried the following :
```
for (n,m) in L:
for i in range(-1,2):
for j in range(-1,2):
if gcd(n+i,m+j)!=1:
print(n,m)
```

which returns any point with $gcd =1$ which is not what I want.
Thanks very much for your help!

I suppose you test only for (i,j) different from (0,0), otherwise (55,21) wouldn't be a solution.

(1275, 1309) work even with (i,j)=(0,0)

I am not quite sure what you mean, since $\gcd(1275,1309)=17$.

I mean that (1275,1309) is a solution for the same problem (with same notations as in my answer) with R=[(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)]