A261782 -id:A261782 - cp's OEIS frontend

A264901 Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.

16, 32, 64, 64, 128, 128, 128, 243, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2744, 4096, 4096, 4096, 4096, 6561, 6561, 6561, 6561, 8192, 8192, 8192, 8192, 8192, 8192

Offset: 1

Anatoly E. Voevudko, Nov 28 2015

We do not distinguish between the representations C^z = A^x + B^y and C^z = B^y + A^x.

This sequence is based on the type of equation involved in Beal's conjecture.

			128 = 64 + 64 ==> 2^7 = 2^6 + 2^6 = 2^6 + 4^3 = 4^3 + 4^3 (but not 4^3 + 2^6).

Anatoly E. Voevudko, Table of n, a(n) for n = 1..615
American Mathematical Society, Beal Prize
Anatoly E. Voevudko, Description of all powers in b245713
Anatoly E. Voevudko, Description of all powers in b261782
Anatoly E. Voevudko, Description of all powers in b264901
Wikipedia, Beal's conjecture

Cf. A245713, A261782.

b264901(lim)=
{my(Lc=List(1),Lb=List(),La=Lb,czn,lan,lbn,lcn,lim2=logint(lim,2),lim3);
for(z=3,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));)); lbn=#Lb; listsort(Lb);
for(i=1,lbn, print(i," ",Lb[i]))}
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);}

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

Original entry on oeis.org

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

Anatoly E. Voevudko, Dec 14 2015

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

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

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

Cf. A000290, A245713, A261782, A264901, A265732.

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

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

Original entry on oeis.org

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

Anatoly E. Voevudko, Dec 14 2015

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.

			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).

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

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

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

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A264901 Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.

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

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PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI