A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.

8, 9, 16, 16, 25, 25, 32, 32, 32, 36, 36, 64, 64, 64, 81, 81, 100, 100, 100, 125, 125, 128, 128, 128, 128, 128, 128, 144, 144, 169, 196, 225, 225, 225, 225, 243, 256, 256, 256, 289, 289, 289, 324, 324, 324, 343, 400, 400, 400, 441, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 576

Offset: 1

Author

Anatoly E. Voevudko, Dec 14 2015

Keywords

• nonn
• easy

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.

Examples

			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).
		

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

Crossrefs

Cf. A000290, A245713, A261782, A264901, A265731.

Programs

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