20200226, 14:35  #1 
Mar 2018
2·269 Posts 
Two variable functions
223*34^x+5879*y(y2)*17^2+10=f(x,y)
23*34^x+5879*y(y2)*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? Last fiddled with by enzocreti on 20200226 at 14:46 
20200227, 04:52  #2 
Aug 2006
5,987 Posts 
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.
Code:
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) 
20200227, 06:13  #3  
Mar 2018
2×269 Posts 
Quote:
Among the solutions there are k=69660, 92020 and 541456 which are the exponents of pg primes multiple of 86. Is it a chance? 

20200227, 07:17  #4 
Aug 2006
1763_{16} Posts 
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.

20200227, 08:15  #5  
Mar 2018
2×269 Posts 
...
Quote:
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? Last fiddled with by enzocreti on 20200227 at 09:00 

20200227, 10:01  #6  
Mar 2018
21A_{16} Posts 
...
Quote:
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 

20200228, 02:21  #7 
Aug 2006
5,987 Posts 

20200228, 08:50  #8 
Mar 2018
2·269 Posts 
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