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以äžã§ã¯ãèªç¶æ°ã¯ 0 {\displaystyle 0} ãå«ããšãã a {\displaystyle a} ã b {\displaystyle b} ãå²ãåãããš(ã€ãŸã b = c a {\displaystyle b=ca} ãšãªãèªç¶æ° c {\displaystyle c} ãååšããããš)ã a ⣠b {\displaystyle a\mid b} ãšè¡šãã
åå gcd : N n â N ; ( a 1 , ... , a n ) ⊠d {\displaystyle \gcd \colon \mathbb {N} ^{n}\to \mathbb {N} ;(a_{1},\dots ,a_{n})\mapsto d} ã
ããã«å®ããã d {\displaystyle d} ã a 1 , ... , a n {\displaystyle a_{1},\dots ,a_{n}} ã®æå€§å
¬çŽæ°ãšããã gcd ( a 1 , ... , a n ) {\displaystyle \gcd(a_{1},\dots ,a_{n})} ã ( a 1 , ... , a n ) {\displaystyle (a_{1},\dots ,a_{n})} ãšè¡šãã gcd ( a 1 , ... , a n ) = 1 {\displaystyle \gcd(a_{1},\dots ,a_{n})=1} ãæãç«ã€ããšã a 1 , ... , a n {\displaystyle a_{1},\dots ,a_{n}} ãäºãã«çŽ ã§ãããšèšãã
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ããšãã° 333 {\displaystyle 333} ãš 57 {\displaystyle 57} ã®æå€§å
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a 1 , ... , a n {\displaystyle a_{1},\dots ,a_{n}} ã«å¯Ÿã㊠( a 1 , ... , a n ) = ( gcd ( a 1 , ... , a n ) ) = ( a 1 ) + ⯠+ ( a n ) {\displaystyle (a_{1},\dots ,a_{n})=(\gcd(a_{1},\dots ,a_{n}))=(a_{1})+\cdots +(a_{n})} ãæãç«ã¡ããã匷ãå€é
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ã§ããã®ã§ãåççã«ã¯ gcd ( a , b ) {\displaystyle \gcd(a,b)} ã ( a , b ) {\displaystyle (a,b)} ãšæžãããšãæ£åœåãããŠãããšè§£éã§ãããç¹ã«ã空éåã®çæããã€ãã¢ã«ãé¶ã€ãã¢ã«ã§ããããšããã空éåã®æå€§å
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åŒæéã§æ€èšŒã§ããã以äžãããçŽ æ°å€å®åé¡ã¯NPã«å±ãããåææ°å€å®åé¡ã®çµæãšäœµãããš NP â© co-NP {\displaystyle {\textsf {NP}}\cap {\textsf {co-NP}}} ã«å±ããããšãèšããããªããçŸåšã§ã¯çŽ æ°å€å®åé¡ã¯ P ã«å±ããããšã蚌æãããŠããã
NP ã«å±ãããã P ã«å±ããããšã蚌æãããŠããªãåé¡ã®å€ã㯠NPå®å
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åŒæéã§è§£ããåé¡ã®ã¯ã©ã¹ã§ãããå®çŸ©ããæããã« P â NP {\displaystyle {\textsf {P}}\subseteq {\textsf {NP}}} ã ãã P â NP {\displaystyle {\textsf {P}}\neq {\textsf {NP}}} ãã©ããã¯æªè§£æ±ºåé¡ã§ãã(Pâ NPäºæ³)ããŸã co-NP ã¯ã NP ã®è£åé¡ã®ã¯ã©ã¹ã§ãããã€ãŸãã
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ãããã¯ã©ã¹ãš NP ã¯å€é
åŒéå±€ãæ§æããã P = Σ 0 P = Î 0 P {\displaystyle {\textsf {P}}=\Sigma _{0}^{\rm {P}}=\Pi _{0}^{\rm {P}}} ãšããé©åã«å®çŸ©ããããšã«ãã£ãŠã
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šã¯ã©ã¹ã®åéåã PH ãšããããã®ãšãã NP = Σ 1 P {\displaystyle {\textsf {NP}}=\Sigma _{1}^{\rm {P}}} ã co-NP = Î 1 P {\displaystyle {\textsf {co-NP}}=\Pi _{1}^{\rm {P}}} ã§ããããã k ã«ã€ã㊠Σ k P = Σ k + 1 P {\displaystyle \Sigma _{k}^{\rm {P}}=\Sigma _{k+1}^{\rm {P}}} ããªã㡠Σ k P = Î k P {\displaystyle \Sigma _{k}^{\rm {P}}=\Pi _{k}^{\rm {P}}} ãšãªãããšããå€é
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ã€ãŸãPâ NPäºæ³ã¯ãå€é
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è¶³å¯èœæ§åé¡ãNPå®å
šã«å±ããããšãç¥ãããŠãã(ã¯ãã¯âã¬ãã³ã®å®ç(è±èªç))ãNPå®å
šåé¡ã®1ã€ã§ã P ã«å±ããããšã蚌æã§ããã° P=NP ãšèšãããããã®ãããªåé¡ã¯èŠã€ãã£ãŠããªãã
NEXPTIMEã¯ãéæ±ºå®æ§ãã¥ãŒãªã³ã°ãã·ã³ã䜿ã£ãŠææ°æéã§è§£ããåé¡ã®ã¯ã©ã¹ã§ãããæééå±€å®ç(è±èªç)ãã NP â NEXPTIME {\displaystyle {\textsf {NP}}\subsetneq {\textsf {NEXPTIME}}} ãèšããããŸããEXPSPACEã¯ãæ±ºå®æ§ãã¥ãŒãªã³ã°ãã·ã³ã®ææ°ãµã€ãºã®é åã䜿ã£ãŠè§£ããåé¡ã®ã¯ã©ã¹ã§ãããé åéå±€å®ç(è±èªç)ãã NP â EXPSPACE {\displaystyle {\textsf {NP}}\subsetneq {\textsf {EXPSPACE}}} ãèšããã
å€é
åŒæéæ€èšŒå¯èœãšããåŽé¢ãã NP ãèãããšã察話蚌æãšã¿ãªãããšãã§ãããAMã¯(P ã BPP ã«æ¡åŒµããããã«) NP ãã確ççãªæåãèš±ãããã«ããã¯ã©ã¹ã§ãããPCPå®ç(è±èªç)ã¯ã NP = PCP ( O ( log n ) , O ( 1 ) ) {\displaystyle {\textsf {NP}}={\textsf {PCP}}(O(\log n),O(1))} ã瀺ããŠããã
| false | false | false |
https://ja.wikipedia.org/wiki/NP
|
9,912 |
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æå°å
¬åæ°(ãããããããã°ããããè±: least common multiple)ãšã¯ã 0 {\displaystyle 0} ã§ã¯ãªãè€æ°ã®æŽæ°ã®å
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2ã€ä»¥äžã®æŽæ° a 1 , ... , a n {\displaystyle a_{1},\ldots ,a_{n}} ã®æå°å
¬åæ°ãšã¯ã a 1 , ... , a n {\displaystyle a_{1},\ldots ,a_{n}} ã®å
¬åæ°ã®ãã¡æå°ã®æ£æŽæ°ã§ããã
ã€ãŸãã a 1 , ... , a n {\displaystyle a_{1},\ldots ,a_{n}} ããçŽ æ° (prime) p ãçšããŠ
a j = ε j â p : p r i m e p e p ( j ) ( e p ( j ) ⥠0 , ε j = ± 1 ) {\displaystyle a_{j}=\varepsilon _{j}\prod _{p:\,\mathrm {prime} }p^{e_{p}(j)}\ \ \ (e_{p}(j)\geq 0,\ \ \varepsilon _{j}=\pm 1)}
ãšçŽ å æ°åè§£ãããšãã a 1 , ... , a n {\displaystyle a_{1},\ldots ,a_{n}} ã®æå°å
¬åæ°ã¯
â p : p r i m e p max { e p ( 1 ) , ... , e p ( n ) } {\displaystyle \prod _{p:\,\mathrm {prime} }p^{\max\{e_{p}(1),\ldots ,e_{p}(n)\}}}
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a , b , c , ⯠, z {\displaystyle a,b,c,\cdots ,z} ã®æå°å
¬åæ°ã l {\displaystyle l} ãšãã. a , b , c , ⯠, z {\displaystyle a,b,c,\cdots ,z} ã®äžè¬ã®å
¬åæ°ã m {\displaystyle m} ãšã, m = q l + r , ( 0 < r < l ) {\displaystyle m=ql+r,\quad (0<r<l)} ãšçœ®ãã å€åœ¢ã㊠r = m â q l {\displaystyle r=m-ql} ...1 1å³èŸºã¯ m {\displaystyle m} 㯠a , b , c , ⯠, z {\displaystyle a,b,c,\cdots ,z} ã®å
¬åæ°ã l {\displaystyle l} ãåãã a , b , c , ⯠, z {\displaystyle a,b,c,\cdots ,z} ã®å
¬åæ°ã ãã£ãŠ1ã®å·ŠèŸº r {\displaystyle r} 㯠a , b , c , ⯠, z {\displaystyle a,b,c,\cdots ,z} ã®å
¬åæ°ã«ãªãã ããã 0 < r < l {\displaystyle 0<r<l} ãšãªããæå°å
¬åæ° l {\displaystyle l} ãããäžè¬å
¬åæ° r {\displaystyle r} ãå°ããççŸ. ããªãã¡ r = 0 {\displaystyle r=0} ããã£ãŠå
¬åæ° m = q l {\displaystyle m=ql} ã§ããæå°å
¬åæ°ã®åæ°ãšãªã£ãŠãã.(蚌æçµ)
æ£æŽæ° a , b {\displaystyle a,\ b} ã«å¯ŸããŠã a {\displaystyle a} ãš b {\displaystyle b} ã®æå€§å
¬çŽæ° g c d ( a , b ) {\displaystyle \mathrm {gcd} (a,\ b)} ãšæå°å
¬åæ° l c m ( a , b ) {\displaystyle \mathrm {lcm} (a,\ b)} ãšã®éã«ã¯
g c d ( a , b ) â
l c m ( a , b ) = a b {\displaystyle \mathrm {gcd} (a,\ b)\cdot \mathrm {lcm} (a,\ b)=ab}
ãšããé¢ä¿ãããã
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¬åæ°ãšãããäŸãã°ã x 3 â x {\displaystyle x^{3}-x} ãš x 3 + x 2 â x â 1 {\displaystyle x^{3}+x^{2}-x-1} ã®æå°å
¬åæ°ã¯ x ( x + 1 ) 2 ( x â 1 ) {\displaystyle x(x+1)^{2}(x-1)} ã§ããã
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| false | false | false |
https://ja.wikipedia.org/wiki/%E6%9C%80%E5%B0%8F%E5%85%AC%E5%80%8D%E6%95%B0
|
9,913 |
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RFC 1242ã®3.17ã§ã¯ãããã®æ©åšã«ãã£ãŠéä¿¡ãã¬ãŒã ãæå€±ããªãæå€§ã¬ãŒãããšå®çŸ©ãããŠãããããŒã¿ã¹ããªãŒã äžã®1ã€ã®ãã¬ãŒã ãæ¬ ãããšããŠããäžäœãããã³ã«ã®ã¿ã€ã ã¢ãŠããåŸ
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https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%AB%E3%83%BC%E3%83%97%E3%83%83%E3%83%88
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9,928 |
ãããã¬
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ãããã¬(è±: debugger)ãšã¯ããããã°äœæ¥ãæ¯æŽããã³ã³ãã¥ãŒã¿ããã°ã©ã ã®ããšãå©çšè
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https://ja.wikipedia.org/wiki/%E3%83%87%E3%83%90%E3%83%83%E3%82%AC
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9,929 |
ZIP
|
ZIP, Zip, zip
| true | false | false |
https://ja.wikipedia.org/wiki/ZIP
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9,930 |
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https://ja.wikipedia.org/wiki/%E3%82%A4%E3%83%B3%E3%82%B9%E3%82%BF%E3%83%B3%E3%82%B9
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9,966 |
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Robocode
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NECã°ã«ãŒã
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https://ja.wikipedia.org/wiki/%E6%83%85%E5%A0%B1%E7%90%86%E8%AB%96
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