# Canonical Decomp

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• June 18th 2010, 08:14 PM
dwsmith
Canonical Decomp
$2^{27}+1=(2^9)^3+1^3=(2^9+1)(2^{18}-2^9+1)$ $=(2^3+1)(2^6-2^3+1)(2^{18}-2^9+1)$

What next?
• June 18th 2010, 08:27 PM
roninpro
It looks like you can write $2^3+1=(2+1)(2^2-2+1)$.

Is this what you are aiming for?
• June 18th 2010, 08:29 PM
dwsmith
No, this section:
$(2^6-2^3+1)(2^{18}-2^9+1)$
• June 18th 2010, 08:30 PM
roninpro
As polynomials, you can't factor those anymore.
• June 18th 2010, 09:00 PM
dwsmith
I have broken it down to $3^3*19*(2^9*7*73+1)$ but it isn't fully decomp.

What to do next?
• June 19th 2010, 12:01 AM
Opalg
$2^n$ is congruent to $\pm1$ mod 3, according as n is even or odd. So $2^{18}-2^9+1$ is a multiple of 3. In fact it is equal to 3*87211, and 87211 is prime, as you can check here.