On the inverse problem relative to dynamics of the <Emphasis Type="Italic">w</Emphasis> function期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the inverse problem relative to dynamics of the w function

Authors:

ChaoHua Jia

Affiliation:

(1) Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Abstract:

Let P be the set of prime numbers and P(n) denote the largest prime factor of integer n > 1. Write

$begin{gathered} C_3 = { p_1 p_2 p_3 :p_i in mathcal{P}(i = 1,2,3),p_i ne p_j (i ne j)} , hfill B_3 = { p_1 p_2 p_3 :p_i in mathcal{P}(i = 1,2,3),p_1 = p_2 or p_1 = p_3 or p_2 = p_3 , but not p_1 = p_2 = p_3 } . hfill end{gathered}$

For n = p ₁ p ₂ p ₃ ∈ C ₃ ∪ B ₃, we define the w function by

$$
w(n) = P(p_1 + p_2 )P(p_1 + p_3 )P(p_2 + p_3 ).
$$

If there is m ≠ S ⊂ C ₃ ∪ B ₃ such that w(m) = n, then we call m S-parent of n. We shall prove that there are infinitely many elements of C ₃ which have enough C ₃-parents and that there are infinitely many elements of B ₃ which have enough C ₃-parents. We shall also prove that there are infinitely many elements of B ₃ which have enough B ₃-parents.

Keywords:

dynamics prime number w function

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