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interests / rec.puzzles / Re: Factorize: 2016 and ( 81 ^ 3 + 4 )

SubjectAuthor
* Factorize: 2016 and ( 81 ^ 3 + 4 )henh...@gmail.com
`* Re: Factorize: 2016 and ( 81 ^ 3 + 4 )Edward Murphy
 `* Re: Factorize: 2016 and ( 81 ^ 3 + 4 )henh...@gmail.com
  `* Re: Factorize: 2016 and ( 81 ^ 3 + 4 )Edward Murphy
   `- Re: Factorize: 2016 and ( 81 ^ 3 + 4 )henh...@gmail.com

1
Factorize: 2016 and ( 81 ^ 3 + 4 )

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Subject: Factorize: 2016 and ( 81 ^ 3 + 4 )
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Wed, 29 Jun 2022 22:23 UTC

1. Factorize: 2016

2. Factorize: ( 81 ^ 3 + 4 )

pls wait a few days before posting answers or hints.

Re: Factorize: 2016 and ( 81 ^ 3 + 4 )

<t9j66h$3m2$2@gioia.aioe.org>

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From: emurph...@zoho.com (Edward Murphy)
Newsgroups: rec.puzzles
Subject: Re: Factorize: 2016 and ( 81 ^ 3 + 4 )
Date: Wed, 29 Jun 2022 20:44:16 -0700
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 by: Edward Murphy - Thu, 30 Jun 2022 03:44 UTC

On 6/29/2022 3:23 PM, henh...@gmail.com wrote:

> 1. Factorize: 2016
>
> 2. Factorize: ( 81 ^ 3 + 4 )
>
>
> pls wait a few days before posting answers or hints.

How is this not just routine arithmetic? Do you have some
particular shortcut in mind?

rot13(
1. gjb rkc svir gvzrf guerr rkc gjb gvzrf frira
2. svir rkc bar uhaqerq svsgl frira rkc fvk uhaqerq friragl frira
)

Re: Factorize: 2016 and ( 81 ^ 3 + 4 )

<0750cc18-923b-42db-a0c1-1b2c7785f09cn@googlegroups.com>

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Subject: Re: Factorize: 2016 and ( 81 ^ 3 + 4 )
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Thu, 30 Jun 2022 04:04 UTC

On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
> On 6/29/2022 3:23 PM, henh...@gmail.com wrote:
>
> > 1. Factorize: 2016
> >
> > 2. Factorize: ( 81 ^ 3 + 4 )

is the second one SO routine ????

for the 1st one, i meant to write :

1b. Factorize: 2009


> >
> > pls wait a few days before posting answers or hints.
> How is this not just routine arithmetic? Do you have some
> particular shortcut in mind?
>
> rot13(
> 1. gjb rkc svir gvzrf guerr rkc gjb gvzrf frira
> 2. svir rkc bar uhaqerq svsgl frira rkc fvk uhaqerq friragl frira
> )

Re: Factorize: 2016 and ( 81 ^ 3 + 4 )

<t9stfc$1s4r$1@gioia.aioe.org>

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From: emurph...@zoho.com (Edward Murphy)
Newsgroups: rec.puzzles
Subject: Re: Factorize: 2016 and ( 81 ^ 3 + 4 )
Date: Sun, 3 Jul 2022 13:16:43 -0700
Organization: Aioe.org NNTP Server
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 by: Edward Murphy - Sun, 3 Jul 2022 20:16 UTC

On 6/29/2022 9:04 PM, henh...@gmail.com wrote:

> On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
>> On 6/29/2022 3:23 PM, henh...@gmail.com wrote:
>>
>>> 1. Factorize: 2016
>>>
>>> 2. Factorize: ( 81 ^ 3 + 4 )
>
>
> is the second one SO routine ????

It works out to 531445, which obviously has a factor of 5. That leaves
106289, whose factors (157 and 677) are less obvious.

In light of factorizing 2009, we can also try to express 531445 as a
difference of squares:

81 = 3^4, s 81^3 = 3^12 = (3^6)^2 = 729^2. The number in question
equals:

* 729^2 + 4 (given), so for a difference of squares, we need to
start with the square of an integer > 729.

* 730^2 - 1455 (not a square)

* 731^2 - 2916
= 731^2 - 54^2
= (731 + 54) * (731 - 54)
= 785 * 677

From there, by inspection 785 has a factor of 5, and then a bit of
trial and error confirms that 5 * 157 * 677 can't be broken down
any further.

> for the 1st one, i meant to write :
>
> 1b. Factorize: 2009

Trial and error quickly leads to 7 * 7 * 41

If you notice that it's close to 2025 = 45^2, and then notice that the
difference is 16 = 4^2, then you can do
2009 = 45^2 - 4^2
= (45 + 4) * (45 - 4)
= 49 * 41

Re: Factorize: 2016 and ( 81 ^ 3 + 4 )

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Subject: Re: Factorize: 2016 and ( 81 ^ 3 + 4 )
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sun, 3 Jul 2022 20:31 UTC

On Sunday, July 3, 2022 at 1:16:50 PM UTC-7, Edward Murphy wrote:
> On 6/29/2022 9:04 PM, henh...@gmail.com wrote:
>
> > On Wednesday, June 29, 2022 at 8:44:19 PM UTC-7, Edward Murphy wrote:
> >> On 6/29/2022 3:23 PM, henh...@gmail.com wrote:
> >>
> >>> 1. Factorize: 2016
> >>>
> >>> 2. Factorize: ( 81 ^ 3 + 4 )
> >
> >
> > is the second one SO routine ????

> It works out to 531445, which obviously has a factor of 5. That leaves
> 106289, whose factors (157 and 677) are less obvious.
>

thakns... the formula i had in mind was ...

The [ Sophie Germain Identity ]

a^4 + 4 b^4 = (a^2 + 2b^2 + 2ab) (a^2 + 2b^2 - 2ab)

1
server_pubkey.txt

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