You may consult the following paper by Christian Elsholtz & Terence Tao: https://terrytao.wordpress.com/tag/erdos-straus-conjecture/

A prime $p$ is of ET-type $I$ if there are natural numbers $\ x\ y\ z\ $ such that $\ x\ $, but not $\ y\ $ nor $\ z,\ $ is divisible by $\ p\ $, and

$$ \frac 4p = \frac 1x+\frac 1y+\frac 1z $$

A prime $p$ is of ET-type $II$ if there are natural numbers $\ x\ y\ z\ $ such that both $y$ and $z$, but not $x$, are divisible by $p$, and the above Erdös-Straus equation holds.

Are the ET-type $I$ primes superfluos? -- i.e.

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**QUESTION**:   Is every ET-type $I$ prime of the ET-type $II$ as well?

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You may consult the following paper by Christian Elsholtz & Terence Tao: https://terrytao.wordpress.com/tag/erdos-straus-conjecture/

A natural solution $\ (p\ x\ y\ z)\ $ of Erdös-Straus equation $$ \frac 4p = \frac 1x+\frac 1y+\frac 1z $$ is of ET-type $I$ if $\ x\ $, but not $\ y\ $ nor $\ z,\ $ is divisible by $\ p\ $.

A natural solution $\ (p\ x\ y\ z)\ $ of the same Erdös-Straus equation is of ET-type $II$ if both $y$ and $z$, but not $x$, are divisible by $p$

Are the ET-type $I$ solutions superfluos? -- i.e.

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**QUESTION**:   Is every prime $p$ represented by an ET-type $I$ solution also represented by an ET-type $II$ as well?

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