The diophantine eq. $x^4 +y^4 +1=z^2$ 
This question is an exact duplicate of the question
Does the equation $x^4+y^4+1=z^2$ have a non-trivial solution?
posted by Tito Piezas III on math.stackexchange.com.

The background of this question is this: Fermat proved that the equation, $$x^4 +y^4=z^2$$
has no solution in the positive integers.  If we consider the near-miss, $$x^4 +y^4-1=z^2$$
then this has plenty (in fact, an infinity, as it can be solved by a Pell equation). But J. Cullen, by exhaustive search, found that the other near-miss, $$x^4 +y^4 +1=z^2$$
has none with $0 < x,y < 10^6$  .
Does the third equation really have none at all, or are the solutions just enormous?
 A: Didn't find any solutions with $1 \leq y \leq 7.9 \cdot 10^7$ and no restriction on $x,z$ in about 17 hours on 1 core.
Here is the search:
Per several discussions and arguments $\mod {20}$ both $x,y$ are divisible by $10$.
Rewrite as $$ y^4 + 1 = z^2 - x^4 $$
The RHS is a difference of two squares with the additional restriction the second square must be fourth power, so the algorithm uses a single loop, write $(10y_1)^4+1$ as a difference of two squares (in all possible ways) and checks for fourth power.
A: Let take a look to the special case:
$$
y=ax
$$
where $a$ is a fixed nonzero integer.  When $a=1$ was already observed by Max Alexseyev that there is no solution.
Since
$$
x^4+y^4+(x+y)^4=2(x^2+xy+y^2)^2
$$
the equation $x^4+y^4+1=z^2$ can then be written:
$$
X^2 -DY^4 =1
$$
with
$$
X=z, Y=x,  D = a^4+1.
$$
(Or, more simply, we get this also directly from the original equation...).
Observe now that 
the equation
$$
a^4+1=w^2
$$
has no integer solution $w$.
See e.g.,  Corollary in page 17 of Mordell's Diophantine Equations.
Then it follows from the paper of Togbe et al. below that there are at most $2$ positive solutions
$x,z$ of the equation.
Togbe, A.; Voutier, P. M.; Walsh, P. G.(3-OTTW)
Solving a family of Thue equations with an application to the equation $x^2-Dy^4=1$.
Acta Arith. 120 (2005), no. 1, 39–58.
11D59 (11D25)
Let $D$ be a positive nonsquare integer. The authors study the Diophantine equation $X^2-DY^4=1$ in positive integers $X$ and $Y$ and refine a theorem of W. Ljunggren [Skr. Norske Vid.-Akad. Oslo I 1936, no. 12, 1--73; Zbl 0016.00802]. Let $(T_1,U_1)$ be the smallest integer solution to the Pell equation $X^2-DY^2=1$. For $k\ge 1$, let $T_k+U_k\sqrt D={(T_1+U_1\sqrt D)^k}$ represent all positive integer solutions to the Pell equation. The authors prove:
   There are at most two positive integer solutions $(X,Y)$ to the equation $X^2-DY^4=1$. If two solutions $Y_1 <Y_2$ exist, then $Y_1^2=U_1$ and $Y_2^2=U_2$, except only if $D=1785$ or $D=16\cdot 1785$, in which case $Y_1^2=U_1$ and $Y_2^2=U_4$. If only one positive integer solution $(X,Y)$ exists, then $Y^2=U_l$ where $U_1=lv^2$ for some squarefree integer $l$, and either $l=1$, $l=2$, or $l=p$ for some prime $p\equiv 3\pmod 4$.
   The problem is reduced to solving the family of Thue equations $x^4+4tx^3y-6tx^2y^2-4t^2xy^3+t^2y^4=t_0^2$, where $t_0$ divides $t$ and $t_0\le \sqrt t$, for a positive integer $t$. However, it is not required to solve this family completely, but only for solutions whose quotient $x/y$ is near to $\beta^{(3)}$ or $\beta^{(4)}$, where $\beta^{(j)}$, $j=1,\dots,4$, denote the roots of the univariate polynomial corresponding to the Thue equation in a particular order defined in the paper. For these two roots, an effective measure of irrationality can be proved by Thue's hypergeometric method.
Reviewed by Clemens Heuberger
A: This is not an answer but just a probabilistic (i.e. heuristic) argument which is too long for a comment.
I want to argue that the equation $x^4+y^4+1=z^2$ is likely to have non-trivial solutions and that the fact that there are none with $y<10^8$ isn't exactly a good evidence for lack of solutions. First, looking modulo 4 and 5 we notice that $x, y$ must both be divisible by 10. Also note that modulo other primes there doesn't seem to be any unusual obstructions, i.e. roughly half of the values of $\{x^4+y^4+1; x, y \in \mathbb{Z}/p\mathbb{Z}\}$ are quadratic residues modulo $p$.
We then write our equation as $10^4(x^4+y^4)+1=z^2$. A random number of order $n$ has probability roughly $\tfrac{1}{2\sqrt{n}}$ of being a square but here we know that it is a square modulo $10^4$, so the conditional probability is roughly $4/\sqrt{n}$. So for fixed $y$ the expected number of $x$ which would make $10^4(x^4+y^4)+1$ a perfect square is roughly
$$
\sum_{x=1}^\infty \frac{1}{25\sqrt{x^4+y^4}}\sim \frac{2}{27y},
$$
the last comparison is asymptotic for large $y$ (the exact constant is, of course, not $2/27$) but the value of the sum is somewhat smaller for small $y$.
The sum $\sum_{y=1}^\infty \frac{2}{27y}$ diverges (which suggests there are infinitely many solutions) but if you sum only over $y\leq 10^8$ the sum is still only roughly $1.4$
