The Diophantine equation $x_1^6+x_2^6+x_3^6=z^2$ where exactly one $(x_i)\equiv 0{\pmod 7}$.

Above equation shown below:

$x^6+y^6+z^6=w^2$

There are two more numerical solutions for $(x,y,z)< 5000$ $(x,y,z,w)=(2043,2184,2518,20883327517)$

$(x,y,z,w)=(3087,3404,4482,102604114673)$

Above solutions including numerical solution shown by "OP" $(x,y,z,w)=(140,213,390,60163597)$

have been arrived at through elliptical curve method by Seiji Tomita on his web site shown below:

  http://www.maroon.dti.ne.jp/fermat/eindex.html

Click on the above link & select 'Computional number theory" &

check out his articles # 166 and #167

Equation: $x^6+y^6+z^6=w^2$ ----(1)

Regarding the subsequent request by @OldPeter yesterday, please see below:

Refer to Andrew Bremner & M. Ulas 2011 paper in the International journal of number theory, pages 2018-2090, vol. 8, No 07, having title $ (x^a±y^b±z^c±w^d=0)$

The paper includes additional numerical solutions to equation (1) above:

$(x, y, z, w )$

$694, 945, 1308, 2414891825$

$42, 873, 3596, 46505412377$

$792, 3759, 5038, 138465240337$

$1515, 3262, 5160, 141747483853$

$2975, 4950, 7902, 508783710817$

$4410, 5463, 8270, 594854319097$

$5340, 6626, 9765, 987341285501$

$1689, 10528, 14886, 3498954949801$

$588, 8224, 26097, 17782152244433$

$834, 17094, 21373, 10966834991269$

$1182, 14644, 24597, 15209227541197$

Here is a near-solution. If you have one primitive solution to,

$$(2a)^\color{red}2+b^6+c^6 = (2d)^2$$ you can find an infinite more using the identity,

$$(a x^6 - d x^6 + a y^6 + d y^6)^\color{red}2 + (b x y)^6 + (c x y)^6= (a x^6 - d x^6 - a y^6 - d y^6)^2$$

for arbitrary $x,y$.

Example: Given,

$$(81^3)^2+42^6+100^6 = 1134865^2$$

Hence $a=81^3/2$ and $d = 1134865/2$. Using $x= 2,\,y=1$, then,

$$18476415^2 + 84^6 + 200^6 = 20142721^2$$

and infinitely many more where at least one of the $x,y$ is odd.

P.S. It tantalizes there may be a similar identity if $\color{red}2$ is raised to $6$.

here is the correct parametrization of primitive Pythagorean quadruples