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In the page on the Erdős-Straus conjecture, the result is conjectured to be true for all $n>1$:

$$(4/n)-((1/x)+(1/y)+(1/z))=0⇔nxy+nxz+nyz-4xyz=0$$

My question is about this generalized version:

$$(d/n)-((1/x)+(1/y)+(1/z))=0⇔nxy+nxz+nyz-dxyz=0$$

where $d$ is a positive integer.

(1) Can some one give me a counterexample for certain value of $d$.

(2) Why the presence of $4$ in the equation makes strong believes that the result is true.

John2000
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1 Answers1

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Note that we have $$\frac{d}{n}-[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}]=\frac{d}{n}-[\frac{x+y}{xy}+\frac{1}{z}]=\frac{d}{n}-[\frac{xy+xz+yz}{xyz}]=0$$ $$\implies nxy+nyz+nxz -dxyz=0$$

Note that the Erdős–Straus Conjecture talks about expressing the fraction $\frac{4}{n}$ as a sum of three distinct fractions with numerator $1$ for $n\geq 2$.

For the second part, see the Generalisations section in the same Wiki page.

bof
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