How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Let $n$ and $p$ be positive integers greater than $1$, with $p$ being a prime. Show that if $n$ divides $p-1$ and $p$ divides $n^3-1$, then $4p-3$ is a perfect square.

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" _i="1" _address="0.1.0.2.1" hover_enabled="0"]Note that, $p$ divides either $n-1$ or $n^2+n+1$. Using inequalities, show that $p$ has to be greater than $n-1$. Hence $p|n^2+n+1$. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" _i="2" _address="0.1.0.2.2" hover_enabled="0"]Write $n^2+n+1=kp$ for some integer $k$. Using the fact that $n|p-1$, show that the integer $k$ has residue 1 modulo $n$. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" _i="3" _address="0.1.0.2.3" hover_enabled="0"]Note that, if $p\neq n^2+n+1$, then $k$ has to be greater than or equal to $n+1$. Show that this leads to a contradiction. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" _i="4" _address="0.1.0.2.4" hover_enabled="0"]As $p$ has residue 1 modulo $n$, we have $p\ge n+1$. This means that $n^2+n+1=kp\ge (n+1)(n+1)=n^2+2n+1$. This is clearly a contradiction. Hence $p=n^2+n+1$. Therefore $4p-3=(2n+1)^2$. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.26.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]