A245713 -id:A245713 - cp's OEIS frontend

A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

16, 32, 64, 128, 243, 256, 512, 1024, 2048, 2744, 4096, 6561, 8192, 16384, 32768, 65536, 131072, 177147, 185193, 262144, 474552, 524288, 614656, 810000, 941192, 1048576, 1124864, 1419857, 1500625, 2097152, 3241792, 4194304

Offset: 1

Anatoly E. Voevudko, Aug 31 2015

nonn

Beal's conjecture states that A, B, and C have a common prime factor.

Theorem. If A, B are odd and x, y are even, Beal's conjecture has no counterexample. Proof: Let D be odd, D > 1 and let w be even, w > 2. Then D^w == 9 (mod 24) while D == 0 (mod 3); otherwise, D^w == 1 (mod 24) (trivial). Any even C^z == {0; 8; 16} (mod 24): if C == 0 (mod 3), C^z == 0 (mod 24); if C == 1 (mod 3), C^z == 16 (mod 24); if C == 2 (mod 3), C^z == 8 (mod 24), while z is odd, and C^z == 16 (mod 24), while z is even (trivial). But C^z == (x'+y') (mod 24) where A^x = x' (mod 24), B^y = y' (mod 24); since (x'+y') = {2; 10; 18}, C^z == {2; 10; 18} (mod 24), which cannot be a counterexample to Beal's conjecture. - Sergey Pavlov, May 08 2021

			2^3 + 2^3 = 2^4 = 16, so 16 is in the sequence.

Anatoly E. Voevudko and Charles R Greathouse IV, Table of n, a(n) for n = 1..1229 (first 196 terms from Voevudko)
American Mathematical Society, Beal Prize
Anatoly E. Voevudko, Description of all powers in b245713
Anatoly E. Voevudko, Description of all powers in b261782
Wikipedia, Beal's conjecture

Subsequence of A076467.

Cf. A245713.

is(n)=if(ispower(n)<3, return(0)); for(x=3,logint((n+1)\2,2), for(A=2,sqrtnint(n,x), if(ispower(n-A^x)>2, return(1)))); 0 \\ Charles R Greathouse IV, Sep 03 2015

list(lim)=my(v=List(),u=v,t); for(z=3,logint(lim\=1,2), for(C=2,sqrtnint(lim,z), listput(v,C^z))); v=Set(v); for(i=1,#v, for(j=i,#v, t=v[i]+v[j]; if(t>lim, break); if(setsearch(v,t), listput(u,t)))); Set(u) \\ Charles R Greathouse IV, Sep 03 2015

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

Original entry on oeis.org

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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A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

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

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Examples

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PARI