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수학동아 - 폴리매스
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함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐] &$%&)&*
GUN.007 2019.11.11 02:32 조회 735
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

n은 홀수고, a와 b는 정수일 때

n=a^(2)+/-b^(2)

이 모든 홀수 n에 대해서 언제나 성립함을 증명해라.

(단, a, b는 같아도 상관이 없다.)

이 문제 어떠셨나요?

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  •  
    디듀우 Lv.7 2019.11.11 02:37

    모든 홀수 n에 대해 이것을 만족하는 정수 a, b가 존재함을 증명하라는 건가요?

    댓글 작성하기 좋아요0 댓글수1
    •  
      GUN.007 Lv.11 2019.11.11 02:40

      넵

      좋아요0
  •  
    유한의끝도못본남자 Lv.7 2019.11.11 02:54

    n이홀수라는것은 일단 1부터출발한다 1에 2칸씩 뛰어가면 홀수가나온다

    (-a/b)^2이므로  -는 소거하여도 상관이없다

    (a/b)^2이 홀수가될수있다는것을 증명하면된다

    일단 홀수는 2로 나누어져서는안된다

    수학적귀납법을유도하자

    n이 1이라고가정하면 2/2가되어참이다

    n+2 = 3 따라서 루트 3가되어야한다 하지만 루트3는 무한소수이므로 성립되지않는다

     

    이문제 자체가 틀린것같은데요 .....

    댓글 작성하기 좋아요0 댓글수1
    •  
      GUN.007 Lv.11 2019.11.11 03:04 비밀댓글
      비밀 댓글이 등록 되었습니다!
  •  
    21세기오일러 Lv.11 2019.11.11 04:24

    반증, a가 4고 b가 2일때 

    댓글 작성하기 좋아요0 댓글수1
    •  
      GUN.007 Lv.11 2019.11.11 04:42 비밀댓글
      비밀 댓글이 등록 되었습니다!
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