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$(a+b+c\cdots)\neq(a^{2}+b^{2}+c^{2}\cdots)$ given all distinct values for the variables?

When I came across this topic, it made me curious as to explore other possibilities, as here, what other two equations can XenoGraff use that have an equal number of variables, but have only one unique solution?

Anonym, who stated that this might be possible if it considered cubes, not squares, could be right, but if this isn't possible, then what are the possible other two equations that can be formed such that they have an equal number of variables with a unique solution?

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Mach9
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  • To rephrase my question, can I have a fixed number of equations with a unique common solution for as many variables as provided, not necessarily linear? – Mach9 May 02 '12 at 15:47
  • The system $a+b=c$, $ab = 1$, $c=b+2$ has the unique solution $a=2$, $b=1/2$, $c=5/2$. Is this an example of what you're looking for? – Antonio Vargas May 02 '12 at 16:26
  • Uh huh. And what if I were to take two equations $a+b+c+d+e=X, a^{3}+b^{3}+c^{3}+d^{3}+e^{3}=Y$ for example, where X and Y were fixed given values? – Mach9 May 02 '12 at 18:22
  • What if, indeed. What is your question? – Antonio Vargas May 02 '12 at 21:56
  • See the example I gave in response to Anonym's comment. By the way, it is believed, but not proved, that for any $n$ the equation $x^5+y^5=n$ has at most one set ${{x,y}}$ of positive integer solutions. – Gerry Myerson May 03 '12 at 07:14
  • To clarify, you may think of it as a set of some elements (of natural numbers) from which any $k$-subset's $k$ elements can be found through a minimal number of equations, where each element (taken as a number) is represented by variables like $a,b,c\cdots$ and so on. What I need for this purpose are those few (preferably two, three, four, etc.) equations, not necessarily linear. – Mach9 May 03 '12 at 11:42
  • For $k=2$ you just need one equation. From $a+((a+b)(a+b-1)/2)=n$ you can work out $a$ and $b$, uniquely. For example, if $a+((a+b)(a+b-1)/2)=100$ then you must have $a=9$ and $b=5$. – Gerry Myerson May 04 '12 at 13:10
  • Are there more? May I have some external references on this topic? BTW is this and this link of any help? – Mach9 May 08 '12 at 17:15

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