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New Method Said to Solve Key Problem In Math - NewYork Times August 8, 2002 This algorithm is beautiful It's the best result I've heard in over ten years AKS algorithm
PRIMES Is in P : A Breakthrough for "Everyman" 覦 蠖
AKS algorithm Manindra Agrawal, Neeraj Kayal, Nitin Saxena of Indian Institute of Technology る殊語 蠍 4 - 殊, 覈 "PRIMES is in P" 朱 企 9伎讌襴 朱語 15覈 覓瑚蟆 覲企 螳 Jaikumar Radhakrishnan 螻 Vikraman Arvind 螳 豢襯 覲企伎. ろ碁┯(dartmouth) 螻殊レ 襷♀ Carl Pomerance れ 殊 蟆郁骸襯 讀螻, 蠏碁 讀レ朱 碁碁襯 譟一 , NewYork Times 襴 , 朱語 誤磯血 螻糾. 蟲 覲 蟆. 覈, NewYork Times 螳. 蠍, 蠍一ヾ 蟆郁骸 螳 讀覈覯 殊.
Sieve of Eratosthenes N 語 誤蠍 N螻 觜襦 螳 讀, O(N) PRIMES 'リ幻' n 觜語企. N 伎 觜語 log 2 鐃緒申鐃緒申鐃緒申 = 鐃緒申鐃緒申鐃緒申. O(N) = O(2log2 鐃緒申鐃緒申鐃緒申 ) = O(2鐃緒申鐃緒申鐃緒申 ) 讀 讌 觜襦.
PRIMES is in P ? N 語 , 螻 讌 鐃緒申鐃緒申鐃緒申 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申) 襯 襷譟燕 '蟆一' 螻襴讀 譟伎 .
Before August 2002 '螳一' '語覿' '' 覓語螳 覿襴 '襯企 襴' 覈 n, 伎 襦 a 伎 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 螳 焔渚. 覿 焔渚讌 .
Probabilistic Algorithm 1976 Miller and Robin 煙 願碓, 襯襦 襴 襯 4鐃緒申鐃緒申鐃緒申 覲企 螻, O(鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申2 鐃緒申鐃緒申鐃緒申) PRIMES co-RP
Deterministic Algorithm 1983 Adleman, Pomerance, Rumely 襷 企螻, 朱 襯企 襴襯 伎 蟆 2002 蠍一 蟾讌 豕螻 蟆一 螻襴讀 螳 覲旧° (log 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申(log log log 鐃緒申鐃緒申鐃緒申) . (super-polynomial order)
And... 豕蠏 螻襴讀 elliptic curve abelian varieties of high genus 伎 鐃緒申鐃緒申鐃緒申 覯 覦覲 , 蟆一 旧 螻. 覃 旧 旧 覈視 襯 2鐃緒申鐃緒申鐃緒申 PRIMES ZPP
Manindra Agrawal 1991 IITK Computer Science and Engineering 覦 1999. "Primality and identity testing vis Chinese remaindering" Generalization of Fermat's Little Theorem a n 襦 朱, n 企, 蠏碁Μ螻 襷 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 in ring of polynomials [鐃緒申鐃緒申鐃緒申] 一 願鹸 讌襷 蠍 企給.
Two Bachelor's Project (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 企ゼ 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1 襦 襾語襯 伎. r x 襦蠏碁 , 襾語 麹 螻襴讀朱 螳 伎 螻 螳ロ n 朱, a 襦 覈 r 螻 n 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申) a=1 襦 螻螻 r 蟲 譟一 r100, n1010 , r 螻 n 襦 願, 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 (鐃緒申鐃緒申鐃緒申 1)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申) 企, n 願碓 鐃緒申鐃緒申鐃緒申2 1 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 蟆曙 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申3+ 鐃緒申鐃緒申鐃緒申)
Two Bachelor's Project Neeraj Kayal and Nitin Saxena 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 螻 蠍一ヾ ろ語 郁煙 譟一 襴襷 螳れ 葛る, 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 讀覈 2 r = 2, ..., 4鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申2 鐃緒申鐃緒申鐃緒申 朱 螳ロ. 企 覦朱 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申6+ 鐃緒申鐃緒申鐃緒申) 蟆一 螻襴讀 詞 . 伎 2002.04 "Towards a deterministic polynomial time primality test" 朱 企朱 覦
Changing the Viewpoint 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,鐃緒申鐃緒申鐃緒申 r 螻螻 a 螳 覲蟆 覃?
AKS algorithm again.. 1. Decide if n is a power of a natural number. If so, go to step 5. 2. Choose (q,r,s) satisfying the hypotheses of the theorem. 3. For a = 1, . . . , s1 do the following: (i) If a is a divisor of n, go to step 5. (ii) If (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申), go to step 5. 4. n is prime. Done. 5. n is composite. Done
And.. Fermat's Last Theorem Sophie Germain Prime ... 鐃緒申鐃緒申鐃緒申(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申6 鐃緒申鐃緒申鐃緒申) (g(n)) O(g(n) logk g(n)) for some k 豢渚
But.. 讌 れ煙 企朱, 讀 n 覓危襦 螳 觜襯伎襷, れ 一企 覯(512觜) るジ 觜襯 螻襴讀 れ .
谿瑚襭 http://www.ams.org/notices/200305/fea-bornemann.pdf http://www.cse.iitk.ac.in/users/manindra/algebra/primalit y_v6.pdf

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PrimesIsInP

  • 1. New Method Said to Solve Key Problem In Math - NewYork Times August 8, 2002 This algorithm is beautiful It's the best result I've heard in over ten years AKS algorithm
  • 2. PRIMES Is in P : A Breakthrough for "Everyman" 覦 蠖
  • 3. AKS algorithm Manindra Agrawal, Neeraj Kayal, Nitin Saxena of Indian Institute of Technology る殊語 蠍 4 - 殊, 覈 "PRIMES is in P" 朱 企 9伎讌襴 朱語 15覈 覓瑚蟆 覲企 螳 Jaikumar Radhakrishnan 螻 Vikraman Arvind 螳 豢襯 覲企伎. ろ碁┯(dartmouth) 螻殊レ 襷♀ Carl Pomerance れ 殊 蟆郁骸襯 讀螻, 蠏碁 讀レ朱 碁碁襯 譟一 , NewYork Times 襴 , 朱語 誤磯血 螻糾. 蟲 覲 蟆. 覈, NewYork Times 螳. 蠍, 蠍一ヾ 蟆郁骸 螳 讀覈覯 殊.
  • 4. Sieve of Eratosthenes N 語 誤蠍 N螻 觜襦 螳 讀, O(N) PRIMES 'リ幻' n 觜語企. N 伎 觜語 log 2 鐃緒申鐃緒申鐃緒申 = 鐃緒申鐃緒申鐃緒申. O(N) = O(2log2 鐃緒申鐃緒申鐃緒申 ) = O(2鐃緒申鐃緒申鐃緒申 ) 讀 讌 觜襦.
  • 5. PRIMES is in P ? N 語 , 螻 讌 鐃緒申鐃緒申鐃緒申 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申) 襯 襷譟燕 '蟆一' 螻襴讀 譟伎 .
  • 6. Before August 2002 '螳一' '語覿' '' 覓語螳 覿襴 '襯企 襴' 覈 n, 伎 襦 a 伎 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 螳 焔渚. 覿 焔渚讌 .
  • 7. Probabilistic Algorithm 1976 Miller and Robin 煙 願碓, 襯襦 襴 襯 4鐃緒申鐃緒申鐃緒申 覲企 螻, O(鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申2 鐃緒申鐃緒申鐃緒申) PRIMES co-RP
  • 8. Deterministic Algorithm 1983 Adleman, Pomerance, Rumely 襷 企螻, 朱 襯企 襴襯 伎 蟆 2002 蠍一 蟾讌 豕螻 蟆一 螻襴讀 螳 覲旧° (log 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申(log log log 鐃緒申鐃緒申鐃緒申) . (super-polynomial order)
  • 9. And... 豕蠏 螻襴讀 elliptic curve abelian varieties of high genus 伎 鐃緒申鐃緒申鐃緒申 覯 覦覲 , 蟆一 旧 螻. 覃 旧 旧 覈視 襯 2鐃緒申鐃緒申鐃緒申 PRIMES ZPP
  • 10. Manindra Agrawal 1991 IITK Computer Science and Engineering 覦 1999. "Primality and identity testing vis Chinese remaindering" Generalization of Fermat's Little Theorem a n 襦 朱, n 企, 蠏碁Μ螻 襷 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 in ring of polynomials [鐃緒申鐃緒申鐃緒申] 一 願鹸 讌襷 蠍 企給.
  • 11. Two Bachelor's Project (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 企ゼ 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1 襦 襾語襯 伎. r x 襦蠏碁 , 襾語 麹 螻襴讀朱 螳 伎 螻 螳ロ n 朱, a 襦 覈 r 螻 n 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申) a=1 襦 螻螻 r 蟲 譟一 r100, n1010 , r 螻 n 襦 願, 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 (鐃緒申鐃緒申鐃緒申 1)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申) 企, n 願碓 鐃緒申鐃緒申鐃緒申2 1 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 蟆曙 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申3+ 鐃緒申鐃緒申鐃緒申)
  • 12. Two Bachelor's Project Neeraj Kayal and Nitin Saxena 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 螻 蠍一ヾ ろ語 郁煙 譟一 襴襷 螳れ 葛る, 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,1 讀覈 2 r = 2, ..., 4鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申2 鐃緒申鐃緒申鐃緒申 朱 螳ロ. 企 覦朱 O(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申6+ 鐃緒申鐃緒申鐃緒申) 蟆一 螻襴讀 詞 . 伎 2002.04 "Towards a deterministic polynomial time primality test" 朱 企朱 覦
  • 13. Changing the Viewpoint 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申,鐃緒申鐃緒申鐃緒申 r 螻螻 a 螳 覲蟆 覃?
  • 14. AKS algorithm again.. 1. Decide if n is a power of a natural number. If so, go to step 5. 2. Choose (q,r,s) satisfying the hypotheses of the theorem. 3. For a = 1, . . . , s1 do the following: (i) If a is a divisor of n, go to step 5. (ii) If (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 1, 鐃緒申鐃緒申鐃緒申), go to step 5. 4. n is prime. Done. 5. n is composite. Done
  • 15. And.. Fermat's Last Theorem Sophie Germain Prime ... 鐃緒申鐃緒申鐃緒申(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申6 鐃緒申鐃緒申鐃緒申) (g(n)) O(g(n) logk g(n)) for some k 豢渚
  • 16. But.. 讌 れ煙 企朱, 讀 n 覓危襦 螳 觜襯伎襷, れ 一企 覯(512觜) るジ 觜襯 螻襴讀 れ .