Canonical Decomp 2^{27}+1: A Breakdown of the Equation's Components

[Dustinsfl]

$$2^{27}+1=(2^9)^3+1^3=(2^9+1)(2^{18}-2^9+1)=(2^3+1)(2^6-2^3+1)(2^{18}-2^9+1)$$

Now what?

 

[Dick]

$$2^{27}+1=(2^9)^3+1^3=(2^9+1)(2^{18}-2^9+1)=(2^3+1)(2^6-2^3+1)(2^{18}-2^9+1)$$

Now what?

That would really depend a lot on what the question is. Wouldn't it?

 

[Dustinsfl]

That would really depend a lot on what the question is. Wouldn't it?

Canonical Decomp.

 

[Dick]

I give up. What's Canonical Decomp?

 

[Dustinsfl]

I give up. What's Canonical Decomp?

Canonical Decomp of a $\mathbb{Z}^+$ $n$ is of the form $n=p_{1}^{a_1}*p_{2}^{a_2}\dots p_{k}^{a_k}$, where $p_1,\ p_2, \dots \ p_k$ are distinct primes with $p_1,< p_2, < \dots \ <p_k$ and each exponent is a $\mathbb{Z}^+$

 

[Mark44]

2³ + 1 can be factored.

 

[Mark44]

Also, 2⁶ - 2³ + 1 and 2¹⁸ - 2⁹ + 1 are not prime.