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[수학 정보] 매스파이의 소수 정리

매스파이 2021.08.03 01:10 조회 872

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

제가 n번째 소수의 근사값을 구하는 식을 발견했는데요.

$P_n\simeq \sqrt{n^2+2n}\ln n$ (P_n은 n번째 소수. )

이 식의 정확도가 궁금해서 Octave,Wolfram Alpha를 이용해서 정확도를 게산해봤는데요

흠 꽤 정확도가 높더라고요

정확도 계산하는거를 좀 더 발전시키니 다음과 같은 식이 나왔습니다.

$\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{P_n}$

제가 울프럼 알파로 n에 1조를 대입해 봤는데 0.92 정도가 나왔어요.

제 생각에는 $\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{P_n}=1$ 인거 같은데 맞나요? 증명 또는 반증해주세요.

문제 해결 현황

2021.07.31 소수 공식 발견

2021.08.01 Octave로 정확도 계산

2021.08.01 $\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{P_n}$ 의 극한값이 궁금해졌고, $\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{P_n}=1$ 일 것 같다고 추측함(증명은 없습니다. 그냥 '그럴 것 같다' 고 생각해낸 것입니다. )

2021.08.02 게시글 올림

2021.08.04 이 추측을 '매스파이의 소수 정리' 라고 이름붙임.

2021.08.05 Scubed 님이 매스파이의 소수 정리와 관련된 명제를 증명하심.

2021.08.08 토론 게시판에 토론 글 올림.http://www.polymath.co.kr/contents/view/32218

이전글 ☆★깜짝 이벤트★☆ 폴리매스가 '눈에 안띄게' 변했다?

다음글 삼각함수 메들리

추천 2 목록보기

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피카파이 Lv.8 2021.08.03 22:53

이걸 증명하라는 소리는 (저에게는)제타함수를 구하라는 말 같이 들립니다.

댓글 작성하기 좋아요0 댓글수0
매스파이 Lv.8 2021.08.04 23:53

이 추측을 '매스파이의 소수정리(Mathπ's Prime Number Theorem)' 라고 이름 붙이겠습니다.

댓글 작성하기 좋아요1 댓글수1
- SON7 Lv.5 2021.08.05 01:59
  
  55 뭔진 모르겠지만 멋있네요
  
  좋아요0
RTAI Lv.7 2021.08.05 19:22

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Scubed Lv.7 2021.08.06 08:24

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

저 공식을 증명한 건 아니고 이상한 걸 증명했는데 뭐냐면

'매파의 소수정리'가 맞다면 최고차항의 계수가 1인 $f(n)$ 에 대해서 $P_n\geq f(n)$ 을 만족하는 $f(n)$ 의 차수는 2 미만이다.

라는 괴상한 명제인데요, 대충 뭔 느낌이냐면

매파의 소수정리가 맞다면 $P_n\geq n^2-100, P_n\geq n^3-n^2, P_n\geq n^{2.5}-9n$ 같은 부등식이 절대 성립할 수 없다는 것입니다.

증명)

매파의 소수정리가 맞다면 $P_n=\sqrt{n^2+2n}\ln n\geq f(n)$ 이므로

$\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{f(n)}\geq 1$

여기서 로피탈의 정리를 이용하면

$\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{f(n)}=\lim_{n\rightarrow \infty}\frac{\frac{\sqrt{n^2+2n}}{n}+\frac{(n+1)\ln n}{\sqrt{n^2+2n}}}{f'(n)}$ $\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+2n}\ln n}{f(n)}=\lim_{n\rightarrow \infty}\frac{\frac{\sqrt{n^2+2n}}{n}+\frac{(n+1)\ln n}{\sqrt{n^2+2n}}}{f'(n)}=\lim_{n\rightarrow \infty}\frac{\ln n+1}{f'(n)}\geq 1$

근데 여기서 $f'(n)$ 의 차수가 1이상이면 위 부등식이 성립하지 않습니다. 그러므로 $f(n)$ 의 차수는 2미만입니다. 사실 더 줄일 수는 있지만 의미가 없을 것 같네요.

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- 매스파이 Lv.8 2021.08.09 20:45
  
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  $P_n=\sqrt{n^2+2n}\ln n$ 은 아니지 않나요?그렇게 되려면 n이 무한히 커져야 되는데요...
  
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- Scubed Lv.7 2021.08.10 00:16
  
  무한대로 보내면 1이라는 것이 매스파이의 정리이잖아요. 그러면 결국에 n을 무한대로 보내면은 로피탈의 정리를 사용해도 괜찮지 않을까요?
  
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PF＿ Lv.8 2022.03.14 05:29

순간 이거랑 똑같다고 생각했네요 ㅋㅋ(클릭)

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