A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.

11, 46, 52, 117, 142, 198, 236, 286, 415, 488, 524, 549, 621, 666, 835, 873, 908, 970, 1001, 1199, 1388, 1432, 1692, 1757, 1962, 1964, 1971, 2035, 2041, 2366, 2392, 2630, 2655, 2681, 2702, 2815, 2826, 3195, 3421, 3544, 3664, 3715, 4048, 4070, 4097, 4356

Offset: 1

Author

R. J. Mathar, Feb 06 2017

Keywords

• nonn

Comments

If x and y are coprime, so obviously are also (x,z) and (y,z).

The ordered values of the bases of the cubes, z, are a subsequence of (and conjecturally the same as) A008846.

For production purposes we advice to use the parametrized representations (see references).

Examples

			2^2 + 11^2 = 5^3, so 11 is in the sequence.
9^2 + 46^2 = 13^3, so 46 is in the sequence.
47^2 + 52^2 = 17^3, so 52 is in the sequence.
44^2 + 117^2 = 25^2, so 117 is in the sequence.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1172
• Imin Chen, On the equation s^2+y^(2p)=alpha^3, Math. Comp. 77 (262) (2008) 1223-1227.
• Sander R. Dahmen, A refined modular approach to the diophantine equation x^2+y^(2n)=z^3, arXiv:1002.0020 [math.NT] (2010).

Crossrefs

Subsequence of A282093. Cf. A099533.

Programs

Maple

# slow version for demonstration only.
isA282095 := proc(y)
    local x,z3 ;
    for x from 1 to y do
        if igcd(x,y) = 1 then
            z3 := x^2+y^2 ;
            if isA000578(z3) then
                return true ;
            end if;
        end if;
    end do:
    return false ;
end proc:
for y from 1 do
    if isA282095(y) then
        printf("%d,\n",y) ;
    end if;
end do:

Mathematica

okQ[y_] := Module[{x, z3}, For[x=1, xJean-François Alcover, Dec 04 2017, after R. J. Mathar *)

Formula

{y: x^2 + y^2 = z^3; gcd(x,y) = 1; 1 <= x <= y; x, y, z in N}