Message-ID:

It's very inconvenient to be mortal -- you never know when everything may suddenly stop happening.

interests / rec.puzzles / What is (x^2022) modulo (x^2 + 1) ?

What is (x^2022) modulo (x^2 + 1) ?

<608fcdd0-efd8-4bbe-b700-36a63a436543n@googlegroups.com>

https://novabbs.com/interests/article-flat.php?id=517&group=rec.puzzles#517

X-Received: by 2002:ac8:5c91:0:b0:35b:bc2d:527 with SMTP id r17-20020ac85c91000000b0035bbc2d0527mr24301033qta.674.1664314950623;
Tue, 27 Sep 2022 14:42:30 -0700 (PDT)
X-Received: by 2002:a05:6808:23d2:b0:350:b340:6eb5 with SMTP id
bq18-20020a05680823d200b00350b3406eb5mr2912324oib.232.1664314950383; Tue, 27
Sep 2022 14:42:30 -0700 (PDT)
Path: i2pn2.org!i2pn.org!usenet.blueworldhosting.com!feed1.usenet.blueworldhosting.com!peer02.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: rec.puzzles
Date: Tue, 27 Sep 2022 14:42:30 -0700 (PDT)
Injection-Info: google-groups.googlegroups.com; posting-host=2601:648:8600:f450:0:0:0:c05b;
posting-account=YjTkGAoAAAA4_fbAISfvtIqrYbghMeBx
NNTP-Posting-Host: 2601:648:8600:f450:0:0:0:c05b
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <608fcdd0-efd8-4bbe-b700-36a63a436543n@googlegroups.com>
Subject: What is (x^2022) modulo (x^2 + 1) ?
From: henha...@gmail.com (henh...@gmail.com)
Injection-Date: Tue, 27 Sep 2022 21:42:30 +0000
Content-Type: text/plain; charset="UTF-8"
X-Received-Bytes: 1298

by: henh...@gmail.com - Tue, 27 Sep 2022 21:42 UTC

When you divide (x^2022) by (x^2 + 1) , what is the remainder ?

until recently, i'd not encountered problems of this type.

( x^100 ) modulo (x + 1)

( x^100 ) modulo (x^2 + 1)

( x^2022 ) modulo (x^3 + 1)

Re: What is (x^2022) modulo (x^2 + 1) ?

<YhCdnat1k_mb6K7-nZ2dnZfqn_tj4p2d@brightview.co.uk>

copy mid

https://novabbs.com/interests/article-flat.php?id=520&group=rec.puzzles#520

copy link Newsgroups: rec.puzzles

Path: i2pn2.org!i2pn.org!weretis.net!feeder6.news.weretis.net!news.misty.com!border-2.nntp.ord.giganews.com!nntp.giganews.com!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!nntp.brightview.co.uk!news.brightview.co.uk.POSTED!not-for-mail
NNTP-Posting-Date: Tue, 27 Sep 2022 22:11:18 +0000
Subject: Re: What is (x^2022) modulo (x^2 + 1) ?
Newsgroups: rec.puzzles
References: <608fcdd0-efd8-4bbe-b700-36a63a436543n@googlegroups.com>
From: news.dea...@darjeeling.plus.com (Mike Terry)
Date: Tue, 27 Sep 2022 23:11:17 +0100
User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:68.0) Gecko/20100101
Firefox/68.0 SeaMonkey/2.53.12
MIME-Version: 1.0
In-Reply-To: <608fcdd0-efd8-4bbe-b700-36a63a436543n@googlegroups.com>
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <YhCdnat1k_mb6K7-nZ2dnZfqn_tj4p2d@brightview.co.uk>
Lines: 101
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-UJFmKLzPpelXCMHmgW8UUKMV45vgp7P7nX/Tg1Mr1XBeWnQCFyICiI2x+TCau9AMU4KXGaPTekjayfW!9tyL2szJJbSz1satlW+gCKlOQvMb3FkNY8S/zJDUKGTTJ5B4S7TfJSPLWvNIMUMzzZa9pYZwvlBY!jKACjVyUWT9uT0S5AKvquR5THxU=
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40

by: Mike Terry - Tue, 27 Sep 2022 22:11 UTC

On 27/09/2022 22:42, henh...@gmail.com wrote:
>
> When you divide (x^2022) by (x^2 + 1) , what is the remainder ?
>
>
> until recently, i'd not encountered problems of this type.
>
>
> ( x^100 ) modulo (x + 1)
>
> ( x^100 ) modulo (x^2 + 1)
>
> ( x^2022 ) modulo (x^3 + 1)
>

..
..
..
..
..
..
..
..
..
..
..
..
spoiler
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
spoiler
..
..
..
..
..
..
..
..
..
..
..
..

(t - 1) ≡ -1 [mod t]

so (t - 1)^m ≡ (-1)^m ≡ +1 [mod t] (if m even)
≡ -1 [mod t] (if m odd)

This answers all your questions:

> When you divide (x^2022) by (x^2 + 1) , what is the remainder ?

x^2022 = (x^2)^1011, so taking t = x^2 + 1 , m = 1011, the answer is -1
(or x^2, assuming we want a positive remainder)

>
>
> until recently, i'd not encountered problems of this type.
>
>

Similarl approach gives...

> ( x^100 ) modulo (x + 1)>

> ( x^100 ) modulo (x^2 + 1)>

> ( x^2022 ) modulo (x^3 + 1)

Mike.

Subject	Author
What is (x^2022) modulo (x^2 + 1) ?	henh...@gmail.com
Re: What is (x^2022) modulo (x^2 + 1) ?	Mike Terry