Rocksolid Light

Welcome to novaBBS (click a section below)

mail  files  register  newsreader  groups  login

Message-ID:  

The two most beautiful words in the English language are "Cheque Enclosed." -- Dorothy Parker

interests / rec.puzzles / Sums of 2 squares ( == multiple of 6 )

SubjectAuthor
* Sums of 2 squares ( == multiple of 6 )henh...@gmail.com
`* Re: Sums of 2 squares ( == multiple of 6 )Ilan Mayer
 `- Re: Sums of 2 squares ( == multiple of 6 )henh...@gmail.com

1
Sums of 2 squares ( == multiple of 6 )

<4a44dd4d-7909-47d5-952a-e1f07b9c7749n@googlegroups.com>

  copy mid

https://novabbs.com/interests/article-flat.php?id=287&group=rec.puzzles#287

  copy link   Newsgroups: rec.puzzles
X-Received: by 2002:a05:620a:1a12:b0:6af:228c:ef65 with SMTP id bk18-20020a05620a1a1200b006af228cef65mr23045555qkb.680.1657034938404;
Tue, 05 Jul 2022 08:28:58 -0700 (PDT)
X-Received: by 2002:a0c:ca07:0:b0:473:276:9a06 with SMTP id
c7-20020a0cca07000000b0047302769a06mr4226793qvk.56.1657034938215; Tue, 05 Jul
2022 08:28:58 -0700 (PDT)
Path: i2pn2.org!i2pn.org!weretis.net!feeder6.news.weretis.net!news.misty.com!border2.nntp.dca1.giganews.com!nntp.giganews.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: rec.puzzles
Date: Tue, 5 Jul 2022 08:28:57 -0700 (PDT)
Injection-Info: google-groups.googlegroups.com; posting-host=2601:648:8600:bc00:4516:258a:befc:181;
posting-account=YjTkGAoAAAA4_fbAISfvtIqrYbghMeBx
NNTP-Posting-Host: 2601:648:8600:bc00:4516:258a:befc:181
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <4a44dd4d-7909-47d5-952a-e1f07b9c7749n@googlegroups.com>
Subject: Sums of 2 squares ( == multiple of 6 )
From: henha...@gmail.com (henh...@gmail.com)
Injection-Date: Tue, 05 Jul 2022 15:28:58 +0000
Content-Type: text/plain; charset="UTF-8"
Lines: 16
 by: henh...@gmail.com - Tue, 5 Jul 2022 15:28 UTC

Of these numbers ( 2 sets )

Set1= (6, 12, 18, 24, . . . , 150, 156) = (all multiples of 6 under 160)

Set2= (1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248, 1254,
1260, 1266, 1272, 1278, 1284, 1290, 1296, 1302, 1308)

1. which ones are Sums of 2 squares ?

2. while doing these tests by hand, what are some short-cuts ?
(ways of avoiding exhaustive searches)

omg... i almost forgot to ask...
------- pls PLEASE wait a few days before posting answers or hints.

Re: Sums of 2 squares ( == multiple of 6 )

<2ee20c34-914e-43e4-ae0b-4b57d1d14fd0n@googlegroups.com>

  copy mid

https://novabbs.com/interests/article-flat.php?id=302&group=rec.puzzles#302

  copy link   Newsgroups: rec.puzzles
X-Received: by 2002:a05:6214:29ca:b0:472:fa99:100 with SMTP id gh10-20020a05621429ca00b00472fa990100mr10420756qvb.87.1657471487789;
Sun, 10 Jul 2022 09:44:47 -0700 (PDT)
X-Received: by 2002:a05:6214:1cc7:b0:470:5371:26ff with SMTP id
g7-20020a0562141cc700b00470537126ffmr10315809qvd.9.1657471487673; Sun, 10 Jul
2022 09:44:47 -0700 (PDT)
Path: i2pn2.org!i2pn.org!usenet.blueworldhosting.com!feed1.usenet.blueworldhosting.com!peer03.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: Sun, 10 Jul 2022 09:44:47 -0700 (PDT)
In-Reply-To: <4a44dd4d-7909-47d5-952a-e1f07b9c7749n@googlegroups.com>
Injection-Info: google-groups.googlegroups.com; posting-host=2607:fea8:5620:2f8:89c2:a8fa:12fa:cf26;
posting-account=q4H9owkAAAA7qKFiZGLslkOsPXsbTfvM
NNTP-Posting-Host: 2607:fea8:5620:2f8:89c2:a8fa:12fa:cf26
References: <4a44dd4d-7909-47d5-952a-e1f07b9c7749n@googlegroups.com>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <2ee20c34-914e-43e4-ae0b-4b57d1d14fd0n@googlegroups.com>
Subject: Re: Sums of 2 squares ( == multiple of 6 )
From: ilan_no_...@hotmail.com (Ilan Mayer)
Injection-Date: Sun, 10 Jul 2022 16:44:47 +0000
Content-Type: text/plain; charset="UTF-8"
X-Received-Bytes: 2707
 by: Ilan Mayer - Sun, 10 Jul 2022 16:44 UTC

On Tuesday, July 5, 2022 at 11:28:59 AM UTC-4, henh...@gmail.com wrote:
> Of these numbers ( 2 sets )
>
> Set1= (6, 12, 18, 24, . . . , 150, 156) = (all multiples of 6 under 160)
>
> Set2= (1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248, 1254,
> 1260, 1266, 1272, 1278, 1284, 1290, 1296, 1302, 1308)
>
> 1. which ones are Sums of 2 squares ?
>
> 2. while doing these tests by hand, what are some short-cuts ?
> (ways of avoiding exhaustive searches)
>
>
>
> omg... i almost forgot to ask...
> ------- pls PLEASE wait a few days before posting answers or hints.

SPOILER SPACE

..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
If a square n is not divisible by 3, it has the form (3*m+1)^2 or (3*m+2)^2. In either case this means that n has the form 3*m+1.
A sum of two such squares has the form 3*m+2, and thus is not a multiple of 6.
This means the for a multiple of 6 each of the two squares must be the square of a multiple of 3, and both even or both odd.
For set 1 only 9, 36, 81, 144 are candidates.
18 = 9+9, 72 = 36+36 and 90 = 9+81 are the only sums of two squares in this set.
For set 2 only 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296 are candidates.
1224 = 324+900 is the only sum of two squares in this set.

Please reply to ilanlmayer at gmail dot com

__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||

Re: Sums of 2 squares ( == multiple of 6 )

<137e55c7-2be1-4547-9daf-a39bee86dd7fn@googlegroups.com>

  copy mid

https://novabbs.com/interests/article-flat.php?id=311&group=rec.puzzles#311

  copy link   Newsgroups: rec.puzzles
X-Received: by 2002:ad4:4ea2:0:b0:473:6d91:6759 with SMTP id ed2-20020ad44ea2000000b004736d916759mr15816049qvb.102.1657980028324;
Sat, 16 Jul 2022 07:00:28 -0700 (PDT)
X-Received: by 2002:a05:620a:492:b0:6b5:a6ec:b4f3 with SMTP id
18-20020a05620a049200b006b5a6ecb4f3mr12694541qkr.639.1657980028156; Sat, 16
Jul 2022 07:00:28 -0700 (PDT)
Path: i2pn2.org!i2pn.org!weretis.net!feeder6.news.weretis.net!1.us.feeder.erje.net!3.us.feeder.erje.net!feeder.erje.net!border1.nntp.dca1.giganews.com!nntp.giganews.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: rec.puzzles
Date: Sat, 16 Jul 2022 07:00:27 -0700 (PDT)
In-Reply-To: <2ee20c34-914e-43e4-ae0b-4b57d1d14fd0n@googlegroups.com>
Injection-Info: google-groups.googlegroups.com; posting-host=2601:648:8600:bc00:40d0:b8a0:8dae:c21e;
posting-account=YjTkGAoAAAA4_fbAISfvtIqrYbghMeBx
NNTP-Posting-Host: 2601:648:8600:bc00:40d0:b8a0:8dae:c21e
References: <4a44dd4d-7909-47d5-952a-e1f07b9c7749n@googlegroups.com> <2ee20c34-914e-43e4-ae0b-4b57d1d14fd0n@googlegroups.com>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <137e55c7-2be1-4547-9daf-a39bee86dd7fn@googlegroups.com>
Subject: Re: Sums of 2 squares ( == multiple of 6 )
From: henha...@gmail.com (henh...@gmail.com)
Injection-Date: Sat, 16 Jul 2022 14:00:28 +0000
Content-Type: text/plain; charset="UTF-8"
Lines: 70
 by: henh...@gmail.com - Sat, 16 Jul 2022 14:00 UTC

On Sunday, July 10, 2022 at 9:44:48 AM UTC-7, Ilan Mayer wrote:
> On Tuesday, July 5, 2022 at 11:28:59 AM UTC-4, henh...@gmail.com wrote:
> > Of these numbers ( 2 sets )
> >
> > Set1= (6, 12, 18, 24, . . . , 150, 156) = (all multiples of 6 under 160)
> >
> > Set2= (1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248, 1254,
> > 1260, 1266, 1272, 1278, 1284, 1290, 1296, 1302, 1308)
> >
> > 1. which ones are Sums of 2 squares ?
> >
> > 2. while doing these tests by hand, what are some short-cuts ?
> > (ways of avoiding exhaustive searches)
> >
> >
> >
> > omg... i almost forgot to ask...
> > ------- pls PLEASE wait a few days before posting answers or hints.
> SPOILER SPACE
>
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> .
> If a square n is not divisible by 3, it has the form (3*m+1)^2 or (3*m+2)^2. In either case this means that n has the form 3*m+1.
> A sum of two such squares has the form 3*m+2, and thus is not a multiple of 6.
> This means the for a multiple of 6 each of the two squares must be the square of a multiple of 3, and both even or both odd.
> For set 1 only 9, 36, 81, 144 are candidates.
> 18 = 9+9, 72 = 36+36 and 90 = 9+81 are the only sums of two squares in this set.
> For set 2 only 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296 are candidates.
> 1224 = 324+900 is the only sum of two squares in this set.
>

For set 2 -- only 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296 are candidates.
1224 = 324+900 is the only sum of two squares in this set.

> > Set2= (1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248, 1254,
> > 1260, 1266, 1272, 1278, 1284, 1290, 1296, 1302, 1308)

i could check (by hand) only a few numbers in Set2 --- is it practical to expect a pretty lazy person to
check all of these by hand ?

1
server_pubkey.txt

rocksolid light 0.9.8
clearnet tor