Is the number 1280000401 prime or composite?
Validate your answer giving sufficient reasoning.
1280000401 is the value of x^7+x^2+1 evaluated at x=20.
x^7+x^2+1
= (x^7-x^4) + (x^4-x) + (x^2+x+1)
= x^4(x^3-1) + x(x^3-1) + (x^2+x+1)
= x^4(x-1)(x^2+x+1) + x(x-1)(x^2+x+1) + (x^2+x+1)
= (x^2+x+1)*[x^4(x-1) + x(x-1) + 1]
This factorization shows the polynomial x^7+x^2+1 is composite. Then evaluating this at x=20 yields 421*(160000*19+20*19+1) = 421*3040381. Neither factor equals +/-1, so the number 1280000401 is composite.
Edited on August 2, 2023, 9:48 am
Pencil and Paper
