A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

25, 9, 17, 53, 3

Offset: 1

Author

Zhining Yang, Aug 12 2025

Keywords

• nonn
• more

Comments

a(6)>1024.

Examples

			a(3) = 17 because 17^8 = 36840^2 + 273^4 + 20^6 = 82367^2 + 24^4 + 24^6 = 48^2 + 287^4  + 24^6 and for no integer smaller than 17 we have 3 solutions.
		

Links

• Fang, How to find non-trivial positive integer solutions to the Diophantine equation a^2 + b^4 + c^6 = d^8, where gcd(a,b,c) = 1?.

Crossrefs

Cf. A386373.

Programs

Mathematica

f[w_]:=(v={};c=0;w8=w^8;
Do[yy=w8-z^6;Do[xx=yy-y^4;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z}];c++]],{y,Floor[yy^(1/4)]}],{z,Floor[w8^(1/6)]}];{c,w,v});
s=Table[{},5];
For[k=1,k<=60,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=f[k];Print[s[[r]]]]]