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Question Number 71294 by Mohamed Amine Bouguezzoul last updated on 13/Oct/19
solve in Z (1/x)+(1/y)=(1/p) with p∈P
$${solve}\:{in}\:\mathbb{Z}\:\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{1}}{{p}}\:{with}\:{p}\in\mathbb{P} \\ $$
Commented by Rasheed.Sindhi last updated on 13/Oct/19
((x+y)/(xy))=(1/p)  ((xy)/(x+y))=p  xy=(x+y)p  The difference between this question  and  Q#69939 is that there is z∈Z^+   instead of p∈P.I-e this question is  special case of Q#69939.
$$\frac{{x}+{y}}{{xy}}=\frac{\mathrm{1}}{{p}} \\ $$$$\frac{{xy}}{{x}+{y}}={p} \\ $$$${xy}=\left({x}+{y}\right){p} \\ $$$${The}\:{difference}\:{between}\:{this}\:{question} \\ $$$${and}\:\:{Q}#\mathrm{69939}\:{is}\:{that}\:{there}\:{is}\:{z}\in\mathbb{Z}^{+} \\ $$$${instead}\:{of}\:{p}\in\mathbb{P}.{I}-{e}\:{this}\:{question}\:{is} \\ $$$${special}\:{case}\:{of}\:{Q}#\mathrm{69939}. \\ $$
Answered by mr W last updated on 13/Oct/19
x=1+p  y=(1+p)p    examples:  (1/3)+(1/6)=(1/2)  (1/4)+(1/(12))=(1/3)  (1/6)+(1/(30))=(1/5)  ......
$${x}=\mathrm{1}+{p} \\ $$$${y}=\left(\mathrm{1}+{p}\right){p} \\ $$$$ \\ $$$${examples}: \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{6}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{12}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{30}}=\frac{\mathrm{1}}{\mathrm{5}} \\ $$$$…… \\ $$
Commented by Mohamed Amine Bouguezzoul last updated on 13/Oct/19
nice work but there exist more solutions  for example (1/(10))+(1/(10))=(1/5)
$${nice}\:{work}\:{but}\:{there}\:{exist}\:{more}\:{solutions} \\ $$$${for}\:{example}\:\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{10}}=\frac{\mathrm{1}}{\mathrm{5}} \\ $$
Commented by mr W last updated on 13/Oct/19
that′s true sir. please see Q69939  for more.
$${that}'{s}\:{true}\:{sir}.\:{please}\:{see}\:{Q}\mathrm{69939} \\ $$$${for}\:{more}. \\ $$
Commented by Mohamed Amine Bouguezzoul last updated on 13/Oct/19
god bless you sir
$${god}\:{bless}\:{you}\:{sir} \\ $$

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