Given the equation:
$$ x^4 + y^4 = k, $$
where $x$, $y$ and $k$ are distinct non-zero integers, is there any $k$, such that there is more than one solution $\{x, y\}$ for the above equation?
$\endgroup$ 55 Answers
$\begingroup$The biggest number that I found can be expressed as a sum of two 4th powers in more than 1 way is:$2602265219072= 1064^4 + 1072^4=472^4+1264^4$Guys, guess what, I found the other way to represent 11220039255312 as a sum of two 4th powers in 2 different ways:
$11220039255312=1752^4+1158^4= 1536^4+1542^4$
I bet if anyone can find a bigger number than 11220039255312 which can be represented as a sum of two 4th powers in two different ways
$\endgroup$ 7 $\begingroup$Euler showed that $$635318657 = 59^4+158^4 =133^4 + 134^4$$ is the smallest number which can be expressed as the sum of $2$ $4$th positive powers in $2$ different ways.
$\endgroup$ $\begingroup$This is just a generalization of the famous Ramanujan taxicab problem. We are looking for integer solutions of $$ x^4+y^4 = z^4+w^4 $$ or: $$ x^4-z^4 = w^4-y^4 \tag{1}$$ with $\{x,y\}\neq\{z,w\}$. Euler found:
$$635318657 = 133^4 + 134^4 = 158^4 + 59^4\tag{2}$$
and that is the smallest solution.
$\endgroup$ $\begingroup$Here is a list of results by Jarek Wroblewski , the first of which is the famous one by Euler already cited
$\endgroup$ $\begingroup$Since, nobody could find a greater number than 11220039255312 , so, I found it on my own and here it is:$26033514998417=2189^4+1324^4=1784^4+1997^4$
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