0
$\begingroup$

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?

$\endgroup$
5
  • 1
    $\begingroup$ see en.wikipedia.org/wiki/Generalized_taxicab_number $\endgroup$
    – mathlove
    Aug 22 '15 at 13:10
  • $\begingroup$ @mathlove Thanks for that link above. I am a complete no-nothing in number theory so that would serve as a starting point. $\endgroup$
    – balajeerc
    Aug 22 '15 at 13:12
  • $\begingroup$ @mathlove Following the references in the page you linked to above, it appears there is at least one solution: $$ 1584^4 + 594^4 = 1344^4 + 1334^4 = 635318657 $$ (Euler, 1772). Would you like to convert your question to an answer so I can accept it? $\endgroup$
    – balajeerc
    Aug 22 '15 at 13:15
  • $\begingroup$ You have put a superfluous 4 as the last digit of each of your numbers, bala. $\endgroup$ Feb 3 '19 at 6:48
  • $\begingroup$ Should Be Editted. Should say "4th powers" instead of "powers of 4"! $\endgroup$
    – Mike
    Sep 2 at 0:44
2
$\begingroup$

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
  • 2
    $\begingroup$ This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review $\endgroup$
    – Unknown
    Sep 1 at 14:14
  • 2
    $\begingroup$ @Unknown Well, it does answer the question. The question asks if there exists some $k$ which can be expressed as a sum of two 4th powers in more than 1 way. This answer says $k=2602265219072$ works. (One of the other answers gives a link with lots of other solutions, and the solution here is bigger than all of them.) $\endgroup$
    – user1729
    Sep 1 at 14:47
  • 1
    $\begingroup$ @Unknown, please check properly, it does work and it answers the question also $\endgroup$ Sep 1 at 14:48
  • $\begingroup$ I found an even bigger number , that is,11220039255312 which can be expressed as follows: 11220039255312=1536⁴ +1542⁴ But unfortunately, I cannot find the way to represent it in another way but my computer says that it can be represented in 2 ways $\endgroup$ Sep 1 at 14:50
  • 1
    $\begingroup$ @SHOUNAK You could ask this as a new question, and link back to this one. $\endgroup$
    – user1729
    Sep 3 at 7:06
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

Here is a list of results by Jarek Wroblewski http://www.math.uni.wroc.pl/~jwr/422/422-10m.txt, the first of which is the famous one by Euler already cited

$\endgroup$
0
$\begingroup$

Since, nobody could find a greater number than 11220039255312 , so, I found it on my own and here it is: $26033514998417=2189^4+1324^4=1784^4+1997^4$

$\endgroup$
1
  • $\begingroup$ This one also works, and this is the biggest solution I've found yet. $\endgroup$ Sep 3 at 7:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.