- **enzocreti**
(*https://www.mersenneforum.org/forumdisplay.php?f=156*)

- - **Two variable functions**
(*https://www.mersenneforum.org/showthread.php?t=25302*)

Two variable functions223*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? |

[QUOTE=enzocreti;538369]Is it possible to find such k up to 10^6?[/QUOTE]
It is! I find 678 such members. You could, too, with my code below and a copy of [url=https://pari.math.u-bordeaux.fr/]PARI/GP[/url]. 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)[/code] 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. |

[QUOTE=CRGreathouse;538411]It is! I find 678 such members. You could, too, with my code below and a copy of [url=https://pari.math.u-bordeaux.fr/]PARI/GP[/url]. 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)[/code] 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.[/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? |

[QUOTE=enzocreti;538415]Among the solutions there are k=69660, 92020 and 541456 which are the exponents of pg primes multiple of 86.
Is it a chance?[/QUOTE] 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. |

...[QUOTE=CRGreathouse;538419]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.[/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? |

...[QUOTE=CRGreathouse;538419]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.[/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 |

[QUOTE=enzocreti;538433]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[/QUOTE] I mean, znorder(Mod(10,41)) = 5. It's not a rare property. |

...[QUOTE=CRGreathouse;538480]I mean, znorder(Mod(10,41)) = 5. It's not a rare property.[/QUOTE]
yes I don't understand why the k solutions multiple of 43 are all congruent to 344 mod 559 |

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