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ܼ

1

a,b,cˤĤơ夲롣

ax+by=c(*)

(*)򤯤ȤϡŪ˹ͤСľax+by=cʺɸξȤˤ뤳ȤƱǤ롣

1(*)ˤĤơΤ褦˴Ϳ뤳ȤǤ롣

[]d=GCD(a,b)֤ȼΩġ
[1]cdܿǤʤС(*)ʤ
[2]cdܿʤС(*)ʾʤȤ1ġ˻ġ
ޤx=x0,y=y0(*)1ĤβǤС(*)Τ٤Ƥβ򤬼ηͿ롣
x=x_0~+~\frac{b}{d}~t,~y~=~y_0~-~\frac{a}{d}~t(**)
tǤդǤ롣

[](*)äȤСd(*)κդڤΤǡd((*)αդǤ)cڤʤƤϤʤʤ[1]Զ줿Τǡ[1]Ωġ

٤ϡդdcڤȲꤹ롣ΤȤϡ桼åɤθ߽ˡˤäơax+by=dx=x*,y=y*뤳ȤǤ롣

x=x*,y=y*(*)βǤ뤳ȤꡢǤդtФ(**)(*)βǤ뤳ȤϤ˳ΤƷ׻Ǥˡ

դˡx,y(*)βǤȤС(*)ax0+by0=cС

a(x-x0)+b(y-y0)=c-c=0

ȤʤΤǡa(x-x0)=-b(y-y0)Ωġ

ˡd=GCD(a,b)dzС롣

\frac{a}{d}~(x-x_0)~=~-~\frac{b}{d}~(y~-~y_0)(***)

ǡa/db/dϸߤǤʤΤǡ(***)b/dx-x0ڤ뤳Ȥ狼롣Ĥޤꡢtˤäơx-x_0=\frac{b}{d}tɽ롣

(***)ȡy-y_0=-\frac{a}{b}(x-x_0)=-\frac{a}{d}t

2

dȤδط

dФơdͭȤͤ롣

2x2-dy2=0ĤɤȤȴϢƤ롣

򤬻ʤȤ򼨤˹ƱѤ

̿ϴñΤäƤΩĤȤ¿

[̿]xФơʲΩġ

x20 or 1 (mod 4)

[]x4ˡȤ硢0,1,2,3Τɤ줫˹ƱǤ롣x=0,1,2,3ξx24dzä;׻롣

x0123
x20101

ɽ顢x2 (mod 4)ͤ0 or 1ʤ

̿ϡСʿ4ˡȤ23˹Ʊˤʤ뤳ȤϤʤ

[]㤨С171 (mod 4)ʤΤǡ嵭̿17ʿȤΤϴְ㤤Ǥ롣餫17ʿǤϤʤ

嵭̿ʿǤ뤿ˤ0ޤ1ȹƱǤʤФʤʤȤȤǤꡢ0ޤ1ȹƱǤɬʿˤʤȤΤϤǤϤʤ

ˡܤ̿⼨ȤǤ롣

[̿]xФơʲΩġ

x20 or 1 or 4 (mod 8)

[]x8ˡȤ硢0,1,2,3,4,5,6,7Τɤ줫˹ƱǤ롣x=0,1,2,3,4,5,6,7ξx28dzä;׻롣

x01234567
x201410141

ɽ顢x2 (mod 4)ͤ0 or 1ʤ

[̾][1]xʤx=2y(y)ɽȤǤx2=4y2Ǥ롣

ΤȤyʤ顢y2Ǥ뤫顢x2=4y2=4߶=8ܿפǤ롣
ޤyʤ顢y2Ǥ뤫顢x2-4=4(y2-1)=4(-1)=4߶=8ܿפǤ롣

[2]xʤx=2z+1(z)ɽȤǤx2-1=(2z+1)2-1=4z2+4z=4z(z+1)Ȥʤ롣zz+1Ϥɤ餫Ǥ뤿ᡢx2-18ܿǤ롣

ʤȤʿ;

[]ʤȤ򼨤

x2-21y2=38(*)

[]ˡǾ뤿ˡ(*)x,yĤȲꤹ롣

ΤȤ38ǰǤ19ܤơʿ;ιͻԤäƤ

ơ⤷19|yǤС(*)ˤäơ19|x2Ωġ

ȡ19ǿǤ뤫顢19|xȤʤ롣

ǡx,y19dzڤ뤳ȤˤʤäΤǡ(*)κդ192dzڤ롣

(*)αդ192Ǥϳڤʤ顢̷⤹롣

äơy19ȸߤǤǤʤƤϤʤʤ

(*)ξդ19ˡȤƹͤȡΩĤȤ狼롣

x2-21y20 (mod 19)
x2-2y20 (mod 19)(**)ʢ212 (mod 19)

y19dzڤʤΤǡ19ˡȤyεտy-1¸ߤ롣

(**)ξդ(y-1)2ݤȡ롣

(xy-1)22 (mod 19)

219ˡȤʿ;Ǥ뤳Ȥ򼨤Ƥ뤬(2/19)=-12ˡ19ʿ;1ĤˤȤ¤̷⤷Ƥ롣

ʤȤؤα

[]x2-xy-2y2=-1ϲʤȡx,y¸ߤʤȡˤ򼨤

x,y¸ߤȲꤹ롣

ǡ3ˡȤxy0,1,2Τɤ줫˹ƱǤ롣ͤȤ߹碌ˤĤơx2-xy-2y2ͤ׻ơɽ롣

xyx2-xy-2y2
000
01-21
02-81
101
11-21
12-90
2041
210
22-81

äơx2-xy-2y20 or 1Ǥ롣

  • 12 (mod 3)Ǥ뤫顢x2-xy-2y2ϲʤȤ狼롣

[]жx2-xy-2y2=-1жx+y=0x+2y=0ˤʻ̤ʤȤ̣Ƥ롣

ʸ

  • ػϤ褦