수학동아 - 폴리매스

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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함께 풀고 싶은 문제

창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.

[창의 퍼즐] Brocard's problem

뉴턴의 사과 2018.09.13 06:10 조회 832

문제 푼 사람 0명

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

계승은 잘 아시다시피 다음으로 주어집니다.

$n!=\prod_{k=1}^{n}k$

1876년과 1885년에 작성한 논문에서 우리의 주인공 Henri Brocard는

$4!+1=24+1=5^{2}$

$5!+1=120+1=11^{2}$

$7!+1=5040+1=71^{2}$

가 모두 완전제곱수라는 사실에 주목했습니다. 하지만 그는 $n!+1=k^{2}$ 형테의 다른 해를 찾지 못하였고, 이 방정식의 위의 3개 외의 다른 정수해가 존재하는지 물었습니다.

이와는 독립적으로, Ramanujan도 이 문제를 제기하였습니다. 한번 도전해 봅시다

이전글 암호 해독 1탄

다음글 passward맞히기 1탄

이 문제 어떠셨나요?

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math Lv.2 2018.09.15 06:30

아무래도 문제를 먼저 파악을 하는 것이 중요할 것 같습니다. 그대로 코드를 짜서 1부터 100까지 하도록 만들었더니, 30이상에서는 팩토리얼로 인해서 수가 너무 많이 커져서 30~100까지의 모든 수가 n으로 가능하다고 나왔습니다.

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바람개비 Lv.5 2018.09.15 10:11

k가 홀수인 경우가 예시에 나온 경우밖에 없다는 걸 증명하면 되겠네요.

어렵지만 충분히 풀 수 있는 문제라 생각하는데 미해결인가요?

댓글 작성하기 좋아요0 댓글수1
- 뉴턴의 사과 Lv.1 2018.09.16 05:10
  
  넵 미해결입니다
  
  좋아요0

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