Squares that are sum of two squares
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Transcript of Squares that are sum of two squares
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8/8/2019 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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1
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:
:
/