0
大学数学基礎解説
文献あり

A prime-representing function

20
0

W. H. MILLS

A function f(x) is said to be a prime-representing function if ƒ(x) is a prime number for all positive integral values of x. It will be shown that there exists a real number A such that A3x is a prime-representing function, where [R] denotes the greatest integer less than or equal to R.
Let pn denote the nth prime number.

A. E. Ingham has shown that

  1. pn+1pn<Kpn58
    where K is a fixed positive integer.
Lemma

If N is an integer greater than K8 there exists a prime p such that N3<p<(N+1)31 .

Let pn be the greatest prime less than N3. Then
N3<pn+1<pn+Kpn58<N3+KN158<N3+N2<(N+1)31.

Let P0 be a prime greater than K8. Then by the lemma we can construct an infinite sequence of primes, P0,P1,P2,, such that
Pn3<Pn+1<(Pn+1)31.
Let
3.un=Pn3n,vn=(Pn+1)3n.
Then
4.vn>un,un+1=Pn+13n1>Pn3n=un,
5.vn+1=(Pn+1+1)3n1<(Pn+1)3n=vn.
It follows at once that the un form a bounded monotone increasing sequence. Let A=limnun.

A3n is a prime-representing function.

From (4) and (5) it follows that un<A<vn, or Pn<A3n<Pn+1 .

Therefore [A3n]=Pn and [A3n] is a prime-representing function.

参考文献

[1]
W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc. , 1947, 604-604
投稿日:202261
OptHub AI Competition

この記事を高評価した人

高評価したユーザはいません

この記事に送られたバッジ

バッジはありません。
バッチを贈って投稿者を応援しよう

バッチを贈ると投稿者に現金やAmazonのギフトカードが還元されます。

投稿者

tfshhiy
tfshhiy
8
1675

コメント

他の人のコメント

コメントはありません。
読み込み中...
読み込み中