# Expressing an Integer as the Sum of Two Fourth Powers in More than 1 way

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?

• Aug 22 '15 at 13:10
• @mathlove Thanks for that link above. I am a complete no-nothing in number theory so that would serve as a starting point. Aug 22 '15 at 13:12
• @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? Aug 22 '15 at 13:15
• You have put a superfluous 4 as the last digit of each of your numbers, bala. Feb 3 '19 at 6:48
• Should Be Editted. Should say "4th powers" instead of "powers of 4"!
– Mike
Sep 2 at 0:44

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

• 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 Sep 1 at 14:14
• @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.) Sep 1 at 14:47
• @Unknown, please check properly, it does work and it answers the question also Sep 1 at 14:48
• 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 Sep 1 at 14:50
• @SHOUNAK You could ask this as a new question, and link back to this one. Sep 3 at 7:06

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.

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.

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

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

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