Number Theory – Prove the highest power of $ (a ^ 2-ab + b ^ 2) $, which divides $ a ^ p-b ^ p- (a-b) ^ p $ for odd primes $ n $

It's obvious that odd $ n in Bbb N $, $ a ^ n-b ^ n- (a-b) ^ n $ is divisible by $ ab (a-b) $ (With $ n = 1 $ a special case in which $ a ^ n-b ^ n- (a-b) ^ n $ is zero). This can be considered as fact with respect to integers, but is stronger with respect to these polynomials.

It also seems that for prime $ p> 3 $, it is also divisible by $ p $ and there is one more simple factor
$$
Left. (a ^ 2-ab + b ^ 2) right | a ^ p -b ^ p- (a-b) ^ p
$$

and especially if $ p = 6k-1 $ for some $ k in Bbb N $ then $ a ^ p -b ^ p- (a-b) ^ p $ is divisible by $ (a ^ 2-ab + b ^ 2) $ but not through $ (a ^ 2-ab + b ^ 2) ^ 2 $; while if $ p = 6k + 1 $ then $ a ^ p -b ^ p- (a-b) ^ p $ is divisible by $ (a ^ 2-ab + b ^ 2) ^ 2 $,

This property does not depend on $ p $ Be Prime: The remainder of the statement is when it's odd $ n $ is divisible by $ 3 $ then $ a ^ n-b ^ n- (a-b) ^ n $ is not divisible by $ (a ^ 2-ab + b ^ 2) $,

I want to prove that to everyone $ k in Bbb N $

$$
a ^ {6k + 1} -b ^ {6k + 1} – (ab) ^ {6k + 1} = (6k + 1) ab (ab) (a ^ 2-ab + b ^ 2) ^ 2 P_1 ( a, b)
$$

for a polynomial $ P_1 (a, b) in Bbb Z[a,b]$, and
$$
a ^ {6k + 5} -b ^ {6k + 5} – (ab) ^ {6k + 5} = (6k + 5) ab (ab) (a ^ 2-ab + b ^ 2) P_5 (a, b)
$$

for a polynomial $ P_5 (a, b) in Bbb Z[a,b]$ the self is not a multiple of $ (a ^ 2-ab + b ^ 2) $, and
$$
a ^ {6k + 3} -b ^ {6k + 3} – (a-b) ^ {6k + 3} = (6k + 3) a b (a-b) P_3 (a, b)
$$
from where $ P_3 (a, b) $ is not divisible by $ (a ^ 2-ab + b ^ 2) $,

In trying to prove it, I noticed that $ (a ^ 2-ab + b ^ 2) = (a- omega b) (a – omega ^ 2) b $ from where $ omega $ is a non-trivial cube root of the unit. I had hoped that this would shed a light on why a multiple of $ 3 $ would not have the divisibility, but I can not see how that follows.

Is there a more powerful technique with which the given statements "fall out" or at least are easier to prove?

[Download] The PB Code – Stock Options Course

[Download] The PB Code – Stock Options Course

Hello, my name is Ryan
I will do everything I can to make sure you have an absolutely incredible experience as you go through the PB Code training.
I will be here every step of the way to guide you through the process.
I'll take you from the start, as you can earn a full income with stick Stock.

[Download] The PB Code – Stock Options Course

EndlessClix.com 10% PB First 30000 members get a free upgrade! 0.3 $ PB!

Hi Guys. We're excited to bring EndlessClix.com to you in the pre-launch phase. The official launch phase will take place on the 1st of March. It goes like this. First 30,000 members

When you sign up, you get 2 days of free upgrade membership, 30 Cent balance and 20 cents as an account balance.

Anyone upgrading or buying an ad package or just adding money until we reach 30,000 members will receive it 10% bonus on the purchase balance.More surprises are coming!

We created EndlessClix To offer a different and more stable approach to PTC websites, the vision of a website that will last for years to come. A solid business can only overcome time with your help: the members. We are together STARK!

Thanks again in advance for your participation and support!

Yours sincerely,
EndlessClix team.

Hard drive – Time Machine would like to secure 4.50 PB 266.14 GB

Currently, you can not back up through Time Machine because backups fail with the following error message:

"Time Machine could not complete backing up to Time Machine." The backup hard drive requires 4.50 PB for backup, but only 337.69 GB is available. Select a larger backup disk or shrink the backup by excluding files. "

When I go to Time Machine Options, the estimated size of a full backup is 266.14 GB. The source volume is only 500 GB.

Things I've already tried:

  • Run the First Aid Disk Utility on both the startup disk (the drive to be backed up) and the Time Machine diskette
  • Restart the Mac
  • The Time Machine disk is included in "Excluding Elements of Backups." It is.

Does anyone know why Time Machine could be? solid Overestimation of the size required for the backup?

4.50 PB is 4.500 Terabytes, right?