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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.

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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  • $\begingroup$ math.stackexchange.com/questions/1700130/… $\endgroup$
    – individ
    Commented Oct 20, 2016 at 5:54
  • $\begingroup$ @individ, over there they consider $\ \frac 3n = \ldots.\ $ (here, it's about $\ \frac 4n = \ldots\ $). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Oct 20, 2016 at 6:32
  • $\begingroup$ They consider the general case $\frac{t}{q}$ $\endgroup$
    – individ
    Commented Oct 20, 2016 at 6:34
  • $\begingroup$ @individ, thank you for your link. (If you see a direct connection of your post at StackExchange and my q. above about type I and type II, then, please, let me know). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Oct 20, 2016 at 7:19

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