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

    From henhanna@gmail.com@21:1/5 to All on Tue Jul 5 08:28:57 2022
    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.

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  • From Ilan Mayer@21:1/5 to henh...@gmail.com on Sun Jul 10 09:44:47 2022
    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

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    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
    /__ __\
    ||

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    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From henhanna@gmail.com@21:1/5 to Ilan Mayer on Sat Jul 16 07:00:27 2022
    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

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    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 ?

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