Two variable functions

[enzocreti]

223*34^x+5879*y-(y-2)*17^2+10=f(x,y)
23*34^x+5879*y-(y-2)*17^2+10=g(x,y)




I search integers k>0 such that


either


f(x,y)=k



or



g(x,y)=k


have integer solutions x and y greater than 0.




Is it possible to find such k up to 10^6?

[CRGreathouse]

Originally Posted by enzocreti

Is it possible to find such k up to 10^6?

It is! I find 678 such members. You could, too, with my code below and a copy of PARI/GP. The parameters are pretty extensible in case you'd like to modify things.

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

Up to a billion I find 1370532 or 0.1%. Asymptotically I think the density is about 1.5% (42/2795) but it will take a while to get there.

[enzocreti]

Originally Posted by CRGreathouse

It is! I find 678 such members. You could, too, with my code below and a copy of PARI/GP. The parameters are pretty extensible in case you'd like to modify things.

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

Up to a billion I find 1370532 or 0.1%. Asymptotically I think the density is about 1.5% (42/2795) but it will take a while to get there.

Among the solutions there are k=69660, 92020 and 541456 which are the exponents of pg primes multiple of 86.
Is it a chance?

[CRGreathouse]

Originally Posted by enzocreti

Among the solutions there are k=69660, 92020 and 541456 which are the exponents of pg primes multiple of 86.
Is it a chance?

Probably! You cover a lot of residue classes, and your method isn't clear. Had you specified a method *before* I found the list it would have been more compelling.

[enzocreti]

Originally Posted by CRGreathouse

Probably! You cover a lot of residue classes, and your method isn't clear. Had you specified a method *before* I found the list it would have been more compelling.

All the k's (for k multiple of 43) that are solutions if I am not wrong are congruent to (7^3+1) mod 559.
Why?

[enzocreti]

Originally Posted by CRGreathouse

Probably! You cover a lot of residue classes, and your method isn't clear. Had you specified a method *before* I found the list it would have been more compelling.

92020, 69660 and 541456 are moreover congruent to 10^m mod 41...


I don't know if there are other k's solutions that are congruent to 10^m mod 41

[CRGreathouse]

Originally Posted by enzocreti

92020, 69660 and 541456 are moreover congruent to 10^m mod 41...


I don't know if there are other k's solutions that are congruent to 10^m mod 41

I mean, znorder(Mod(10,41)) = 5. It's not a rare property.

[enzocreti]

Originally Posted by CRGreathouse

I mean, znorder(Mod(10,41)) = 5. It's not a rare property.

yes I don't understand why the k solutions multiple of 43 are all congruent to 344 mod 559