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Question

I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete.

Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$.

Where $\bar{x}=(x_1,x_2,\dots x_n)$ and $\bar{y}=(y_1,y_2,\dots y_n)$ are tuples of non-negative integers.

How long does it take to "decide" whether a given $(\bar{w},r)$ is in the "language" $L$? This is obviously in NP but is it NP-Complete remains to be concluded. Is this language NP-Complete?

Motivations

This problem comes out of my own explorations. I have been exploring solutions on hyperspheres and this is a natural generalization of a hypersphere. Note that $(<2,2,2,2,2,2>, 296675)\in L$ because by Lagrange we know that there exists some $(x_1,x_2,x_3,x_4,x_5,x_6)$ which satisfies $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=296675$. An $n>3$ dimensional hypersphere with radius $\sqrt{r}$, where $r$ is a whole number centered at the origin can always be satisfied by some integer coordinates (Lagrange's sum of 4 squares theorem). This makes the "decision" problem easy. There IS some solution though it might be hard to find.

But what about $x_1^2+x_2^3+x_3^4+x_4^5+x_5^6+x_6^7=296675$?

"Is there a solution to the equation above?" is the same as "Is $(<2,3,4,5,6,7>,296675) \in L$?" in my notation above.

This may be a computationally tricky problem to solve. I know of no guarantee that there must be some $(x_1,x_2,x_3,x_4,x_5,x_6)$ which satisfies the equation. On the otherhand, given an "instance" $(x_1,x_2,x_3,x_4,x_5,x_6)=(1,2,3,4,5,6)$ this problem is easy to verify. "Hard to solve but easy to verify" is a hallmark of NP problems.

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closed as off-topic by quid Jul 26 '18 at 13:11

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  • $\begingroup$ I have added this question here: cstheory.stackexchange.com/questions/40990/… $\endgroup$ – Mason Jun 13 '18 at 23:00
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    $\begingroup$ Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. $\endgroup$ – D.W. Jun 14 '18 at 7:42
  • $\begingroup$ Thanks @D.W. I will be more cautious in the future. I am still getting a feel for which community is interested in what content. What do you suggest to do about the situation now? I think the question was better received cstheory.se but have already used my precious reputation to start a bounty here. I think I would like one of the question to be "closed" or put "on hold" while the other community takes a crack at it. I think it's best if it's "on hold" on Math.SE. and has a link to the CSTheory.SE $\endgroup$ – Mason Jun 14 '18 at 10:14
  • $\begingroup$ At @D.W. I flagged the question asking for moderation. $\endgroup$ – Mason Jun 14 '18 at 10:21
  • $\begingroup$ How does this relate to Hilbert's 10th problem? en.wikipedia.org/wiki/Hilbert%27s_tenth_problem $\endgroup$ – Mason Jun 17 '18 at 2:54