OFFSET
1,1
COMMENTS
We do not distinguish between the equations C^z = A^x + B^y and C^z = B^y + A^x.
This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course with additional restrictions and conditions.
LINKS
Anatoly E. Voevudko, Table of n, a(n) for n = 1..16865
Anatoly E. Voevudko, Description of all powers in b265732
Anatoly E. Voevudko, Description of all powers in b265731
Anatoly E. Voevudko, Description of all powers in b245713
Anatoly E. Voevudko, Description of all powers in b261782
Wikipedia, abc conjecture
Wikipedia, Fermat-Catalan conjecture
EXAMPLE
128 = 64 + 64 ==> 2^7 = 8^2 + 8^2 = 8^2 + 4^3 = 8^2 + 2^6 = 4^3 + 4^3 = 4^3 + 2^6 = 2^6 + 2^6 (but not 4^3 + 8^2, 2^6 + 8^2, 2^6 + 4^3).
PROG
(PARI) A265732(lim, bflag=0)=
{my(Lc=List(1), Lb=List(), La=Lb, czn, lcn, lan, lim2=logint(lim, 2), lim3, k);
for(z=2, lim2, lim3=sqrtnint(lim, z); for(C=2, lim3, listput(Lc, C^z)) );
lcn = #Lc; if(lcn==0, return(-1));
for(i=1, lcn, for(j=i, lcn, czn=Lc[i]+Lc[j]; if(czn>lim, next);
La=findinlista(Lc, czn); lan=#La; if(!lan, next);
for(k=1, lan, listput(Lb, czn)))); lcn=#Lb; listsort(Lb, 0);
if(bflag, for(i=1, lcn, print(i , " ", Lb[i]))); if(!bflag, return(Vec(Lb)));
}
findinlista(list, item, sind=1)={my(ln=#list, Li=List());
if(ln==0||sind<1||sind>ln, return(Li));
for(i=sind, ln, if(list[i]==item, listput(Li, i))); return(Li);
} \\ Anatoly E. Voevudko, Nov 23 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Anatoly E. Voevudko, Dec 14 2015
STATUS
approved