2026/056) Prove that 1280000401 is composite.

 We have 

$1280000401 = 1280000000 + 400 + 1= 20^7 + 20^2+ 1$

This is $x^7 + x^2 + 1$ where x = 20

All the terms have coefficient 1

There is an exponent 7 and and an exponent 2  one if of the form 3n + 2 and another of the form 3n+1

This is divisible by $x^2+x+1$ hence composite

Let us prove the same.

$x^7 + x^2 + 1$ 

$= x^7 -x + x^2 + x + 1$

$=x(x^6-1) + x^2 + x + 1$

$=x(x^3+1)(x^3-1) + (x^2 + x + 1)$

$=x (x^3+1)(x-1)(x^2+x+1)+ x^2 + x + 1$

$=(x^2+x+1) (x(x^3+1)(x-1) + 1$

We have factored the same and 

so the given  number is composite