Intereting Posts

Finding vector $x$ so that $Ax=b$ using Householder reflections.
bilinear form – proof
Suppose $p, p+2, p+4$ are prime numbers. Prove that $p = 3$ not using division algorithm.
Ring of holomorphic functions
Polynomials and Derivatives
Change of Variables in Limits
Recurrence $a_n = \sum_{k=1}^{n-1}a^2_{k}, a_1=1$
find the formula of trinomial expansion
Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}$ into a product of fields
How to solve this ODE?
Is it cheating to use the sign function when sieving for twin primes?
If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $$, then $|f'(1/2)|\le 1/4$
How to prove that $\lim\limits_{n\to\infty} \frac{n!}{n^2}$ diverges to infinity?
Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.
Convergence of square of monotone sequence implies convergence of sequence.

I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out.

By solving I mean fining all solutions and proving there are no more.

Keith Conrad showed how to reduce this equation to a different one which was solved by Fermat in his notes about Fermat descent, other than I have no ideas how to solve it. I tried to do descent on it directly but that seems completely impossible so I am interested in other techniques. Thanks very much.

- Proof that $x^2+4xy+y^2=1$ has infinitely many integer solutions
- Finding all integer solutions of $5^x+7^y=2^z$
- Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$
- Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$
- Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
- Sums of powers being powers of the sum

- How to compute rational or integer points on elliptic curves
- $k^{2}+(k+1)^{2}$ being a perfect square for infinitely many $k$
- Solving: $3^m-2=n^2$
- Solve $x(x+1)=y(y+1)(y^2+2)$ for $x,y$ over the integers
- Does the equation $a^{2} + b^{7} + c^{13} + d^{14} = e^{15}$ have a solution in positive integers
- Erdős-Straus conjecture
- Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet?
- Show that $x(x+1) = y^4+y^3+ay^2+by+c$ has a finite number of positive integral solutions.
- Positive integer solutions to $x^2+y^2+x+y+1=xyz$
- The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

A quick spot of googling found Two fourths and a square.

Look at pattern six: the unique coprime solution is $1^4+1^4=2 \cdot 1^2 .$

- Polynomials having as roots the sum (respectively, the product) of two algebraic elements
- Find thickness of a coin
- Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
- Reducing the form of $2\sum\limits_{j=0}^{n-2}\sum\limits_{k=1}^n {{k+j}\choose{k}}{{2n-j-k-1}\choose{n-k+1}}$.
- Number raised to log expression
- Measure of a set in $$
- Asymptotic behaviour of some series
- Relation between continuity of $f$, $g$ and $f\circ g$
- Proof of Fermat's little theorem using congruence modulo $p$
- Set of zeroes of the derivative of a pathological function
- Why is cross product only defined in 3 and 7 dimensions?
- Prove that $ n < 2^{n}$ for all natural numbers $n$.
- First derivative test
- Approximating the compond interest for a loan
- Solving permutation problem sequentially.