Contemplate the 2 features:
start{align*}
f(x, y) ={}& 223cdot 34^x+5590cdot y +588
g(x, y) ={}& 23cdot 34^x+5590cdot y +588
finish{align*}
My objective was to search out integers $okay$ such that both $f(x, y) =okay$ or $g(x, y) =okay$ have integer options $x, y>0$.

I wrote this Pari code:

liste(lim,mul,base=34,step=5590,offset=588)=my(v=Listing(),X=mul*base); lim=1; whereas(X<lim,forstep(n=X+offset+step,lim,step, listput(v,n)); X*=base); Set(v)
listing(lim)=setunion(liste(lim,23),liste(lim,223))
v=listing(1e6)

Now utilizing this different Pari code:

subset1 = choose((x) -> (x % 43 == 0), v);
subset2 = choose((x) -> (x % 43 == 0) && (x % 559 == 0 || x % 559 == 344), v);
vecsort(subset1) == vecsort(subset2)
> 1

you possibly can test that $okay$‘s are congruent to $0$ $pmod {43}$ and are both congruent to $0$ or $344$ $pmod {559}$.

Why? Is that this provable? The code works on my laptop, if there are issues inform me. So I need to present that if $okay=f(x,y)$ or $okay=g(x,y)$ for some $x,y>0$ and $kequiv0 pmod{43}$, then $kequiv0, 344 pmod{559}$.

Probably this query is expounded to this query:

Possible primes of a selected kind