Squares that are sum of two squares

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    X~qnzax |fn| nza xqm eg |te x~qnzax

    EZ

    Xecq|`eix eg Rp|fndeznx |faezamx a~qn|`ei

    EZ

    Xecq|`eix eg v:

    = p:

    / {:

    XNIHF@\ KN@I

    J)\ahf) 1|f Xamax|az &@igezmn|`ei \ahfiecedp

    In|`einc @ix|`|q|a eg \ahfiecedp Xz`indnz# @il`n

    Amn`c nllzaxx maBxnihf`|)ia|

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    RZAGNHA

    @cc canl peq x|ar-jp-x|ar |e mp g`inc zahoei`id) @g peqcc l`zah|cp xo`r ewaz |e |fa

    zaxqc|x# |f`x t`cc xaam |ee heixr`hqeqx ' peqcc rzejnjcp we|a gez ma |e ja `i

    |fa \er >2 geecx `i |fa tezcl c`x|)

    @ oiet |fn| |f`x leaxi| fecl mqhf x`di`g`hniha &ez mnpja# ieia n| ncc. '

    peql rzejnjcp ja kaaz`id n| ma tfai peq zanl |fa |`|ca# jq| @m ejxaxxal t`|f

    iqmjazx ' @ g`il |f`x `i|azax|`id# tf`hf m`df| xaam |z`w`nc |e peq) @ cewa |fa

    rn||azix |fn| av`x| nmeid iqmjazx)

    Nc|feqdf @wa ncxe lawaceral gezmqcna eg n ce| f`dfaz x`di`g`hniha# @cc ie|zawanc |fam iet)

    @i hnxa eg nip azzez ez `g |fazax nip|f`id @wa m`xxal# rcanxa mn`c ma n|

    maBxnihf`|)ia| )

    Xnihf`| Kn`i

    :7|f

    Nqdqx| :2>2

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    Rn||azix `i iqmjazx nmqxa ma) Gezmqcna hni ja lawaceral gez n cnzdaiqmjaz eg |z`w`nc |f`idx# |f`x ja`id eia eg |fam)

    Niptnpx# @cc jad`i;

    \elnp# `| x|zqho mp m`il |fn| &1:

    = &8:

    / &8

    := &1

    :/ &>:

    :# &:1

    := &4

    :/

    &: / >

    >: = 1v: / :

    :< = 4v8 / 8

    Ca| i nxxqma wncqax gzem >#:#8)

    Nhhezl`idcp# `i|adazx ja`id mqc|`rc`al t`|f i hni ja zarzaxai|al jp &:i / >)

    \fa a~qn|`ei ei |fa z`df| x`la hni ja zarzaxai|al nx

    &:i / >i / i

    = :i&i / >)

    \fqx# |fa x~qnzax geccet`id |f`x rn||azi &ad) 1# >8# :1))|fa`z xqhhaxx`wa

    l`ggazaihax gezm ni N)R) eg hemmei l`ggazaiha :

    ' &:i:

    /

    :i:)

    Xe# |fap nza

    &:i / >:

    / &:i:

    / :i:

    = :

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    nza nctnpx xqm eg x~qnzax _o&:i / >]:

    '

    _o&:i:

    / :i]:

    )

    Tfai o=># i=># ta fnwa &1:

    = &8:

    / i=:# ta fnwa &>8:

    = &1:

    / &>::

    )

    Tfai o=8# i=># ta fnwa &>1:

    = &0:

    / &>::)

    Tfai o=# ta fnwa &:2:

    = &>::

    / &>7:)

    \f`x `x kqx| eia eg |fa |prax eg x~qnzax |fn| nza xqm eg |te e|faz x~qnzax)

    Iet ca|x ceeo n| &6: / &>1: = &>4:) @| leaxi| xn|`xgp |fa njewa zaxqc|)

    Ncxe# ceeo n| &:

    >

    >1 = < v &::->

    Ca| i = >#:#8#

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    Ca|x daiaznc`{a ncc |f`x

    _o&n:v

    i:p

    / j:{

    ]:

    `x n x~qnza |fn| `x xqm eg x~qnzax _o&n:v

    i:p

    j:{

    ]:

    '

    &:onvj

    {i

    p

    :)

    Tfai o = ># v = ># p = ># { = ># n = ># j =># &i: / >: = &i: >: / &:i:)

    Tfai o = ># v = ># p = ># { = ># n = :# j =># &:

    = &:

    /

    v = ># p = ># { = ># n = 8# j =:# &0i:

    / # v = ># p = ># { = ># n = 7i:

    / 0:

    = &>7i:

    0:

    /

    &:# v = ># p = ># { = ># n = 8# j =7:

    = &0i:

    >7:

    /

    &:# v = ># p = ># { = ># n = ># j =:# &i:

    / #

    p = ># { = ># n = ># j = : ' i = 1)

    Ncxe# _o&n:v

    / j:{

    i:p

    ]:

    `x n x~qnza |fn| `x xqm eg x~qnzax _o&n:v

    j:{

    i:p

    ]:

    ' &:onvj

    {i

    p

    :)

    \fa cnx| : zaxqc|x hni ja qxal |e g`il ncc x~qnzax |fn| nza xqm eg |te

    x~qnzax) Nh|qnccp i `x za~q`zal eicp `g peq tni| |e mnoa n gezmqcn|e g`il x~qnzax |fn| geccet |fa xnma rn||azi)

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    IE\A ; Jnxal ei |fa gaaljnho @ zaha`wal# xema raerca fnwa jaai

    fnw`id rzejcamx qilazx|nil`id |fa zaxqc|x)

    Rcanxa ie|a |fn| |faxa zaxqc|x nza n heixa~qaiha eg &v:

    / p:

    := &v

    :-

    p:

    :

    /