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以äžã®ãããªæäœãããŠã¿ãŠãã ããã
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ã決ããŠãã ããã
æèšç®ã§ãããªã p 㯠3 ã 5 ã 7 ãa 㯠2 以äžã§ a*p ã 100 ãè¶
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ã³ã³ãã¥ãŒã¿ã§ããå Žåã¯å¥œããªå€§ããã®æ°ã§ãèªç±ã«ã©ããã
1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ã€ãŸããp ã§å²ããš 1 äœãæ°ã a*p 以äžã®èªç¶æ°ã§ 1 ã€éžãã§ãã ããããšããããšã§ãã
p > 2 ã§ããã°ã
2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ãã£ãã®ãã€ã®äœã 2 ããŒãžã§ã³ã§ãã
p > 3 ã§ããã°ã
3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
äœã 3 ããŒãžã§ã³ã§ãã
以äžäœãã 1 ãã€å¢ãããªããç¹°ãè¿ããŠãäœã (p-1) ããŒãžã§ã³ãŸã§å®è¡ããŠãã ããã
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
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(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
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ãããã¯ã3 ã€ãã4 ã€ãã®çµãäœã£ãŠããããã§ãã
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šãŠã®çµã§åãçŸè±¡ãèµ·ããããšã確èªããŠãã ããã
(4) ãå
šãŠã®çµãã§ã¯ãªããã»ãšãã©å
šãŠã®çµããšèšã£ãã®ã¯ãå®ã¯ a ãš p ãäºãã«çŽ ã ãš 1 ã€ã ãäŸå€ãããããã§ããããŠãããã¯ã©ããªçµã§ãäœãèµ·ããã§ãããïŒ
(5) ãããŸã§ã®å®éšã§ãæåã®æ¡ä»¶ãæºãããã㪠p-1 åçµã®ç·æ°ããp ã®åæ° +1 åããããšãçŽåŸããŠãããããšæããŸãã
ãšããã§ãæåã®æ°ã®éžã³æ¹ãããæãåºããŠããã®ãã㪠p-1 åçµã£ãŠäœéããããã§ããã£ãïŒ
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ããŠãDD++æ§ãæç« ã°ããã§ãªããæ°åŒãã¡ãã°ããŠãæžããŠãã ãããšãã£ãšãããããããªãããããªãããªïŒãšæããŸãã
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æãç® a*p ã¯æžããŠãŸãã
å°äœ k+np ãå
šéšæžããŠãŸãã
åŒãç®ã¯ãæç« äžã« 1 åããç»å ŽããŸããããããããã¡ãã¡æžããã»ããããã§ããïŒ
ããã以å€ãæžããããŠãèšç®ãååšããŸããã
ãããªããŸãããã©ãã§ãééããã®ã§ãããïŒ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ã決ããŠãã ããã
æèšç®ã§ãããªã p 㯠3 ã 5 ã 7 ãa 㯠2 以äžã§ a*p ã 100 ãè¶
ããªããããããããšæããŸãã
ã³ã³ãã¥ãŒã¿ã§ããå Žåã¯å¥œããªå€§ããã®æ°ã§ãèªç±ã«ã©ããã
1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ã€ãŸããp ã§å²ããš 1 äœãæ°ã a*p 以äžã®èªç¶æ°ã§ 1 ã€éžãã§ãã ããããšããããšã§ãã
1+r1p (mod p)â¡1
p > 2 ã§ããã°ã
2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ãã£ãã®ãã€ã®äœã 2 ããŒãžã§ã³ã§ãã
2+r2p (mod p)â¡2
p > 3 ã§ããã°ã
3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
äœã 3 ããŒãžã§ã³ã§ãã
3+r3p (mod p)â¡3
以äžäœãã 1 ãã€å¢ãããªããç¹°ãè¿ããŠãäœã (p-1) ããŒãžã§ã³ãŸã§å®è¡ããŠãã ããã
(p-1)+r(p-1)p (mod p)â¡p-1
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{r1,r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1)}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{r1,r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1),a*p}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1),a*p}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{r2-r1,r3-r1,r4-r1,r5-r1,r6-r1,ã»ã»ã»,r(p-1)-r1,a*p-r1}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
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ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
ã®ãšããã§ããã
éžãã æ°ã¯ rk ãããªããŠãk+rk*p ã®æ¹ã§ãã
å
·äœçãªæ°ã§ãããªããšãããã®ãå°ããé ãã䞊ã¹ãã®ã¯é£ãããšæããŸããã
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{1+r1p,2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1)}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{1+r1p,2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1),a*p}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1),a*p}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{2+r2p-(1+r1p),3+r3p-(1+r1p),4+r4P-(1+r1p),5+r5p-(1+r1p),6+r6p-(1+r1p),ã»ã»ã»,(p-1)+r(p-1)p-(1+r1p),a*p-(1+r1p)}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
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2+r2*p < 1+r1*p
ãšãªãå Žåããããå¿
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ã¯ã¡ã¹ãããããã£ãããã«
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埡åèãŸã§ã«ã
æè³ æ¢è¿ª, ç§åŠå²å
¥é 18äžçŽãšãŒãããã®ååŠç 究 : åŠè
ãã¡ã®äº€æµãšè«äº, ç§åŠå²ç 究, 2014-2015, 53 å·», 272 å·, p. 473-, å
¬éæ¥ 2020/12/14, Online ISSN 2435-0524, Print ISSN 2188-7535, https://doi.org/10.34336/jhsj.53.272_473
, https://www.jstage.jst.go.jp/article/jhsj/53/272/53_473/_article/-char/ja
éäžãŸã§äœæ¥ããŠãããããã¯ã¡ã¹ãããããã®åŸã©ããªã£ãã®ãããããŸããããæ°æ¥çµã¡ãŸããã®ã§çµå±ãããäœã ã£ãã®ããšãããã¿ãã©ã·ãã
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æåã«æ瀺ããæ°ã®éžã³æ¹ã¯å
šéšã§ a^(p-1) éããããŸãã
ãã®ãã¡ãå
šãŠã®éžæ㧠a ã®åæ°ãéžãã { a, 2a, 3a, âŠâŠ, (p-1)a } ãšããçµã¯å¯äžæäœ R ã§èªåèªèº«ã«ãªããŸãã
ïŒa ãš p ãäºãã«çŽ ã®ãšãããã®éžã³æ¹ãå¿
ãå¯èœïŒ
ãããŠæ®ãã® a^(p-1) - 1 åã®çµã¯ãåãæ¡ä»¶ãæºããå¥ã®çµãé ã«å·¡ã£ãŠã2 以äžã® p ã®çŽæ°ãååŸã«èªåèªèº«ã«åž°ã£ãŠããŸãã
ããããp ã¯çŽ æ°ãªã®ã§ãã2 以äžã® p ã®çŽæ°ã㯠p 以å€ã«ãããŸããã
ã€ãŸãããã®æäœ R ã§ãããããšç¹ããé¢ä¿ p å 1 ã°ã«ãŒãã« a^(p-1) - 1 åã®ãã®ããããªãããããªãåããããŸãã
ãã£ãŠãa ã p ãšäºãã«çŽ ã§ããã°ãa^(p-1) - 1 㯠p ã®åæ°ã§ããããšã瀺ããâŠâŠãããšããã»ã©ãã¡ããšæžããŠã¯ããŸãããããªãã»ã©ç¢ºãã«æãç«ã¡ããã ãšèšãããããã®ãªã¢ãã£ãã§ããŸããã
âŠâŠãšããããšãªã®ã§ããã
ã¿ãªããããäœã a^(p-1) åã®ãã®ãçšæããŠããããã 1 ã€ãåãé€ããšãæ®ããæŒããªã p åãã€ã®ã°ã«ãŒãã«ãããããããããªãã®ãæãã€ãããæ¯éæããŠãã ããã
å®éã«ãã£ãŠã¿ããšãç°¡åã«äœãããã«èŠããŠãã¡ããã¡ãé£ããã§ãã
DD++æ§ãããã°ãã¯ã
æšæ¥ã1åç®ãŸã§ããããŸãããïŒïŒïŒã®åé¡ãŸã§ããªãå
ãããããã§ãã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ã決ããŠãã ããã
p=7ãa=5ããšããŸãã
ã1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
1+3p=22
ã2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
2+2p=16
ã3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
3+2p=17
ã4, 4+p, 4+2p, âŠâŠ, 4+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
4+3p=25
ã5, 5+p, 5+2p, âŠâŠ, 5+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
5+2p=19
ã6, 6+p, 6+2p, âŠâŠ, 6+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
6+2p=20
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{16,17,19,20,22,25}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{16,17,19,20,22,25,35}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{17,19,20,22,25,35}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{1,3,4,6,9,19}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
{1,3,4,6,9,19}â¡{1,3,4,6,2,5}(mod p=7)
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
(1)
{1,3,4,6,9,19,35}
{3,4,6,9,19,35}
{2,3,5,8,18,34}
{2,3,5,8,18,34}â¡{2,3,5,1,4,6}(mod p=7)
(2)
{2,3,5,1,4,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(3)
{1,2,3,4,5,6,35}
{2,3,4,5,6,35}
{1,2,3,4,5,34}â¡{1,2,3,4,5,6}(mod p=7)
(4)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(5)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(6)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(7)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
çã
{1,2,3,4,5,6}(mod p=7)
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸã¡ãã£ãŠãŸãïŒ
ã1 åç®ã®çµæã㯠{2,3,5,8,18,34} ã§ããã
ãã(2) ã® 1 åç®ãã€ãŸãå šäœã® 2 åç®ã®çµæã®ããšã§ãã
ïŒ(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
ããããã£ãã€ããã§ãããã»ã»ã»ã»
ééã£ãŠãŸãããïŒ
ïŒ(3) æåã®çµãæ°ãã«äœãçŽããåã³æäœ R ã p åç¹°ãè¿ããŠã¿ãŠãã ããã
ãããã¯ã3 ã€ãã4 ã€ãã®çµãäœã£ãŠããããã§ãã
ã»ãšãã©å
šãŠã®çµã§åãçŸè±¡ãèµ·ããããšã確èªããŠãã ããã
ã§ãããã¯ã©ããªããã ãããšæ¢ãŸã£ãã®ã§ãã
{1,3,4,6,9,19,35} <- æåŸã« 35 ãã€ããïŒæäœ R ã® 1 ã€ãïŒ
{3,4,6,9,19,35} <- 1 ãåãèœãšããŠïŒæäœ R ã® 2 ã€ãïŒ
{2,3,5,8,18,34} <- å
šéšãã 1 ãåŒãããå®æïŒæäœ R ã® 3 ã€ãïŒ
{2,3,5,8,18,34}â¡{2,3,5,1,4,6}(mod p=7) <- äœãããã©ãã©ã確èªããã ã
ããŠãæäœ R ã®å®æåã¯ã©ãã§ãããïŒ
æ¬åœã« {2,3,5,1,4,6} ã§ããïŒ
ãã 1 åæäœ R ã®å®çŸ©ããã¡ããšèªãã§ãã ããã
ïŒå°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
ã§ãããã
ãšãããšã
ïŒ{2,3,5,8,18,34} <- å
šéšãã 1 ãåŒãããå®æïŒæäœ R ã® 3 ã€ãïŒ
ã§ããããïŒ
ã¯ãããããæäœ R ã®çµæã§ãã
ã§ãããã次㯠{2,3,5,8,18,34} ããåºçºã«ãªããŸãã
(1)
{1,3,4,6,9,19,35}
{3,4,6,9,19,35}
{2,3,5,8,18,34}
(2)
{2,3,5,8,18,34}
{3,5,8,18,34,35}
{1,3,6,16,32,33}
(3)
{1,3,6,16,32,33}
{3,6,16,32,33,35}
{2,5,15,31,32,34}
(4)
{2,5,15,31,32,34}
{5,15,31,32,34,35}
{3,13,29,30,32,33}
(5)
{3,13,29,30,32,33}
{13,29,30,32,33,35}
{10,26,27,29,30,32}
(6)
{10,26,27,29,30,32}
{26,27,29,30,32,35}
{16,17,19,20,22,25}
(7)
{16,17,19,20,22,25}
{17,19,20,22,25,35}
{1,3,4,6,9,19}
çã
{1,3,4,6,9,19}
ããã§ããã£ãŠãŸãããïŒ
ã¯ããååã§ãã£ãŠããããšã¯ãã£ãŠããŸãã
ãã ã{16,17,19,20,22,25} ãã {1,3,4,6,9,19} ãäœã£ãã®ã 1 åç®ã§ãããã
> (6)
> {10,26,27,29,30,32}
> {26,27,29,30,32,35}
> {16,17,19,20,22,25}
ã 7 åç®ã§ãããã§ã¹ãããã§ãã