Solving $\tau^4 +1=u^2 +v^2$ over the integers I'm trying to show, that the solutions $t_i$ from Pell's equation $t^2-2s^2=\pm 1$ are not squares (only for the trivial cases). For the even solutions $t_{2i}$ (solutions for the + sign) this is easy, but for the odd solutions $t_{2i+1}$ (solutions for the - sign) this is a little bit harder.
This question leads to the diophantine equation $\tau^4-2s^2=-1$ and the new question, whether it has nontrivial solutions or not.
I do the following approach. First of all I take the equation
$\tau^4+1=u^2+v^2$ (*),
try to find solutions $(u, v, \tau)$ and set in the end $s=u=v$. With
$\begin{align}
u&=u_2 a^2 +u_1 a+u_0 \\
v&=v_2 a^2 +v_1 a+v_0 \\
\tau&=\tau_1 a+\tau_0
\end{align}$
and comparison of the coefficients one gets
$\begin{align}
u&=2a^2 b^2 (b^4 +1)+4ab^3 +1 \\
v&=a^2 (b^8-1)+2ab(b^4 -1)+b^2 \\
\tau&=a(b^4 +1)+b
\end{align}$
with natural numbers $a$ and $b$.
Now the question: are there any other solutions for (*) and if not, why?
Maybe there is also an easy way showing, that the $t_{2i+1}$ are not squares, then I would prefer this proof.
thank you
 A: First, we can find not-all-solutions but  gain insights by re-arranging terms.
$$\tau^4 +1=u^2 +v^2\quad \implies  
\tau^4-u^2 =v^2-1\quad \text{or}\quad  \tau^4 -v^2=u^2-1$$
$$\implies (\tau^2)^2-u^2 =v^2-1\quad \text{or}\quad (\tau^2)^2 -v^2=u^2-1\\
\implies u=v=\pm\sqrt{\tau} \quad\lor\quad \tau=u=v=\pm1$$
but this is only a tiny subset of solutions and empirical data might offer more insights. Let
$$\tau = \sqrt[4]{u^2 +v^2-1}$$
$$\big\{(0,1,0)\big\} \\ 
\big\{(0,0,1), \space  (1,1,1), \space (2,4,1), \space (3,9,1), \space (4,16,1), \space (5,25,1), \space (6,36,1),\space  \cdots\big\}\\ 
\qquad\qquad\qquad\quad\big\{(2,1,4),\space (3,1,9),\space (4,1,16),\space (5,1,25),\space  (6,1,36)\quad \cdots\big\} $$
$$\big\{(8,56,31), \space (8,31,56)\big\}\quad 
\big\{(9,71,39),\space (9,39,71)\big\} \quad 
 \big\{(14,191,44), \space (18,321,44)\big\}\quad
 \big\{(10,76,65), \space (10,65,76)\big\} \quad\cdots $$
We see a pattern in many/most solutions but $u,v$ values of $31, 39, 44, 56, 65,71,76,\cdots $ make "patterns" more elusive, except that $u,v$ are interchangeable in many cases. These numbers were derive from tests of $u \le 3000\land v\le 100$. Beyond that, we do have the complimentary cases of
$(14,44,191), \space (18,44,321).$ What remains is the puzzle of the non-squares.
