Throughout the paper all the variables are positive reals, $(x,y,z)=1$ means that *x*, *y*, *z* are coprime integers, $2|y$ and $(a,b)=1$ means that *a*, *b* are coprime integers and $2|b$.

Equation (1) can be written as (2) where $X={x}^{k}$, $Y={y}^{k}$, $Z={z}^{2}$:

${X}^{2}+{Y}^{2}={Z}^{k}.$(2)

To investigate the integer solutions of (2) under the stated conditions, two equations (3) and (4) are introduced [1, p. 536] such that $(g,h)=1$:

$X+iY={(g+ih)}^{k},$(3)$X-iY={(g-ih)}^{k}.$(4)

Equations (5) and (6) are respectively obtained from (3) and (4) where $\mathrm{tan}H=h/g$, $0<H<\pi /2$:

$X+iY={({g}^{2}+{h}^{2})}^{k/2}[\mathrm{cos}(kH)+i\mathrm{sin}(kH)],$(5)$X-iY={({g}^{2}+{h}^{2})}^{k/2}[\mathrm{cos}(kH)-i\mathrm{sin}(kH)].$(6)

Equation (7) is obtained by multiplying the corresponding sides of (5) and (6):

$({X}^{2}+{Y}^{2})={({g}^{2}+{h}^{2})}^{k}[{\mathrm{cos}}^{2}(kH)+{\mathrm{sin}}^{2}(kH)].$(7)

Since ${\mathrm{cos}}^{2}(kH)+{\mathrm{sin}}^{2}(kH)=1$, equation (8) is obtained from (7):

$({X}^{2}+{Y}^{2})={({g}^{2}+{h}^{2})}^{k}.$(8)

Equation (9) is obtained by comparing (2) and (8):

${z}^{2}={g}^{2}+{h}^{2}.$(9)

Under the assumptions *z*, *g*, and *h* are all integers $>0$, equation (9) represents a right triangle *ZGH* whose sides and area are integers, and *z* is the hypotenuse. Therefore, *ZGH* is a rational right triangle [5]. Equivalently, $(g,h,z)$ is a Pythagorean triple. Consequently, equations (10), (11), (12), and (13) are obtained where $\mathrm{tan}H=h/g$, and $0<H<\pi /2$:

$X={z}^{k}\mathrm{cos}(kH),$(10)$Y={z}^{k}\mathrm{sin}(kH),$(11)$x=z{(\mathrm{cos}kH)}^{1/k},$(12)$y=z{(\mathrm{sin}kH)}^{1/k}.$(13)

By substituting the values of *x* and *y* as obtained from (12) and (13) into (1), equation (14) is obtained by replacing *n* with *k*:

${x}^{2k}+{y}^{2k}={z}^{2k}[{\mathrm{cos}}^{2}(kH)+{\mathrm{sin}}^{2}(kH)].$(14)

Since ${\mathrm{cos}}^{2}(kH)+{\mathrm{sin}}^{2}(kH)=1$, it is concluded that *x* and *y* as obtained in (12) and (13) are indeed the parametric solutions of (1).

From (3) it is noted that $X=Real[{(g+ih)}^{k}]$ and $Y=Imag[{(g+ih)}^{k}]$. Thus one gets (15) and (16), and then (17) and (18), where $j=(k-1)/2$:

$X={\displaystyle \sum _{i=0}^{j}}{(-1)}^{i}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{2i}}\right){g}^{k-2i}{h}^{2i},$(15)$Y={\displaystyle \sum _{i=0}^{j}}{(-1)}^{i}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{2i+1}}\right){g}^{k-2i-1}{h}^{2i+1},$(16)$X=g\left({g}^{k-1}-{C}_{1}{g}^{k-3}{h}^{2}+{C}_{2}{g}^{k-5}{h}^{4}+\mathrm{\cdots}+{(-1)}^{(k-1)/2}k{h}^{k-1}\right),$(17)$Y=sh\left({h}^{k-1}-{C}_{1}{h}^{k-3}{g}^{2}+{C}_{2}{h}^{k-5}{g}^{4}+\mathrm{\cdots}+{(-1)}^{(k-1)/2}k{g}^{k-1}\right),$(18)

where ${C}_{1},{C}_{2},\mathrm{\dots}$ are nonzero integers, each divisible by *k*.
Moreover, $s=+1$ if $k\equiv 1$ (mod 4), and $s=-1$ otherwise. The sign of *s* will influence only the orientation of *X* and *Y* but will have no impact on the integer solutions of (1). Also $(X,Y)=1$ since $(g,h)=1$.

Equations (17) and (18) are rewritten as (19) and (20), respectively, where *Q* and *R* are integers:

$X=gQ,$(19)$Y=hR.$(20)

Since $(g,h)=1$ and $k|h$, one concludes that $(g,Q)=1$ and $(h,R)=k$. Therefore, if *X* and *Y* are *k*-th powers, then *g*, *Q*, *h* and *R* must assume values of the forms $g={u}^{k}$, $Q={w}^{k}$, $h={k}^{rk-1}{v}^{k}$, and $R=k{d}^{k}$, where *u*, *r*, *v*, and *d* are integers $>0$, and *w* is an integer $>1$.

One thus obtains (21) and (22) from (19) and (20):

$\mathrm{tan}kH=Y/X=(h/g)(k{d}^{k}/{w}^{k}),$(21)$\mathrm{tan}kH/\mathrm{tan}H=k{(d/w)}^{k}.$(22)

The impossibility of (22) will imply the impossibility of (1). By expanding $\mathrm{tan}kH$ in terms of $\mathrm{tan}H$ (see [4, p. 111]), one gets (23), where *U* and *V* are given by $U=k{d}^{k}$ and $V={w}^{k}$. It is noted that $\mathrm{tan}H=h/g$ and $(g,h)=1$. One thus gets (24) and (25):

$k{(d/w)}^{k}=U/V,$(23)$k{d}^{k}={h}^{k-1}-{C}_{k-1}{h}^{k-2}+{C}_{k-2}{h}^{k-3}-\mathrm{\cdots}+{C}_{1}{h}^{2}-k{g}^{k-1},$(24)$V={g}^{k-1}-{D}_{k-1}{g}^{k-2}+{D}_{k-2}{g}^{k-3}+\mathrm{\cdots}+{D}_{1}g+{D}_{0}.$(25)

In (24) and (25) the coefficients ${C}_{k-1},{C}_{k-2},\mathrm{\dots}$ and
${D}_{k-1},{D}_{k-2},\mathrm{\dots}$ are all divisible by *k*.

#### Assertion.

*Under the assumption $\mathrm{(}g\mathrm{,}h\mathrm{)}\mathrm{=}\mathrm{1}$ equation (24) can not be satisfied.*

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