平衡素数

平衡素数: balanced prime)は素数であって、1つ前の素数と1つ後の素数の算術平均に等しいものである。代数的に言えば、小さい順に並べたときの n 番目の素数を とすると、 が平衡素数であるとは次が成り立つことである。

平衡素数を小さい順にいくつか列挙すると、

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 オンライン整数列大辞典の数列 A006562.

例えば、53は16番目の素数である。15番目と17番目の素数である4759足す106になり、その半分は53なので、53は平衡素数である。

仮に1を素数であると考えれば、それに応じて2が最初の平衡素数と考えられる。なぜならば

だからである。平衡素数は無限に多く存在すると予想されている。

等差数列における3つの連続した素数は、CPAP-3 と呼ばれることがある。平衡素数は定義によって CPAP-3 の2番目の素数である。2014年現在、CPAP-3として知られている最も大きなものは 10546 桁のものであり、David Broadhurst によって発見された[1]

n の値は知られていない。

関連した概念

素数が、その1つ前の素数と1つ後の素数の算術平均よりも大きい場合、強素数英語版(strong prime)と呼ばれる。逆に小さい場合は、弱素数(weak prime)と呼ばれる。

位数 n の平衡素数

位数 n の平衡素数(balanced prime of order n)とは、素数であって、それに近い上下 n 個ずつの素数の算術平均に等しいようなものである。代数的には、k 番目の素数 が平衡素数であるとは、次が成り立つことである。

前述の素数は位数 1 の平衡素数である。他の位数は 2 オンライン整数列大辞典の数列 A082077、3 オンライン整数列大辞典の数列 A082078、4 オンライン整数列大辞典の数列 A082079 で見られる。

脚注

  1. ^ The Largest Known CPAP's. Retrieved on 2014-06-13.

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