A265731 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, without multiplicity.

8, 9, 16, 25, 32, 36, 64, 81, 100, 125, 128, 144, 169, 196, 225, 243, 256, 289, 324, 343, 400, 441, 512, 576, 625, 676, 784, 841, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1521, 1600, 1681, 1728, 1764, 1849, 2025, 2048, 2197, 2304, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125

Offset: 1

Author

Anatoly E. Voevudko, Dec 14 2015

Keywords

• nonn
• easy

Comments

This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course, with additional restrictions and conditions.

Examples

			8 = 2^3 = 2^2 + 2^2; 9 = 3^2 = 1^3 + 2^3; 16 = 4^2 = 2^3 + 2^3; etc.
		

Links

• Anatoly E. Voevudko, Table of n, a(n) for n = 1..7253
• 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, A265732.

Programs

PARI

A265731(lim,bflag=0)={my(Lcz=List(1),Lb=List(),czn,lczn,lbn,lim2=logint(lim, 2),lim3);
for(z=2, lim2, lim3=sqrtnint(lim, z); for(C=2, lim3, listput(Lcz, C^z)));
Lcz=Set(Lcz); lczn = #Lcz; if(lczn==0,return(-1));
for(i=1, lczn, for(j=i, lczn, czn=Lcz[i]+Lcz[j]; if(czn>lim, break);
if(setsearch(Lcz, czn), listput(Lb, czn)))); listsort(Lb,1);  lbn=#Lb;
if(bflag, for(i=1,lbn,print(i , " ", Lb[i]))); if(!bflag,return(Vec(Lb))); }
\\ Anatoly E. Voevudko, Nov 23 2015