## 3 to the Power of (p-1) is not Congruent to 1 Mod P Cubed if P is Congruent to 1 Mod 6

**Authors:** Ramaswamy Krishnan

If\quad f(a)\quad =\quad 1\quad -\quad { a }^{ p }\quad -\quad { (1-a) }^{ p }\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad and\quad even\quad if\quad { a }^{ 2 }-a+1\quad \ncong \quad 0\quad mod(p)\\ it\quad is\quad prooved\quad that\quad f({ a }_{ r })\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad .Then\quad using\quad the\quad fact\quad that\quad if\\ { 3 }^{ p-1 }\quad \equiv \quad 1\quad mod({ p }^{ 3 })\quad ,\quad { a }^{ 2 }+a+1\quad \equiv \quad 0\quad mod({ p }^{ 3 })\quad is\quad also\quad a\quad solution\quad to\quad \\ f(a)\quad \equiv \quad 0\quad mod({ p }^{ 3 }).

**Comments:** 2 Pages.

**Download:** **PDF**

### Submission history

[v1] 2017-08-17 05:25:20

**Unique-IP document downloads:** 13 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*