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[March for reflection.|.Maynard] degree of digit

已有 423 次阅读 2022-7-20 19:02 |个人分类:牛津大学|系统分类:科研笔记

[注:下文是群邮件的内容,标题是原有的。内容是学习一篇数学文章的笔记。]

["Terms of awareness /use" folded below] On going is to read a paper of primes to increase generic understanding on mathematics.[blog]

Is it possible to alter a digit of a prime to enter another prime?

    ♘   7        5

 

    ♗   2        3

Story - It appears a pleasant day.

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

Recall: properties of primes can be recast as problems about their digits.

---- By the last note, it appears possible to acquire a prime by altering one digit of the given one.

---- Consider a prime of two digits.

---- There are 21 of them.

---- Given one of them, say, 23.

---- One can acquire another prime by altering the first digit to be 1, 4, 5, 7 or 8.

---- That's 5 "gates".

---- Alternatively, one can change the second digit to be 9.

---- There is only one "gate" for the second digit of 23.

---- Question: for any prime, is it possible to alter any single digit of the prime to reach another prime?

---- Or, is it possible to find a prime whose one of the digits cannot serve as a "gate" to another prime?

---- The positive answer to the first question implies each digit at least covers two primes of the same size.

---- By viewing the table of primes smaller than 100, one can find a positive answer for the second question.

---- That answer is 97, denying the first question.

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Definition (degree of digit). Given a prime of k digits. We say the ith digit of the prime is of degree θ, if this digit covers θ primes of k digits.

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Example 1. (primes of one digit). Any digit of a prime of one digit is of degree 4. These primes are 2, 3, 5, 7.

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Example 2. The prime 97. The first digit (9) of the prime is of degree 5, covering 5 primes 17, 37, 47, 67 and 97. The second digit (7) of the prime is of degree one, covering the prime 97 itself.

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Example 3. ("connection" of primes). Take a prime, say, 43. One may draw a graph of connection of primes of 2 digits.

11 --  13  -- 17 -- 19 

  |       |       |        |

  |     23   ----|---- 29

  |       |       |        |

31  --- |---- 37        |

  |       |       |        |

41   -- 43 -- 47       |

  |       |       |        |

  |      53 ----|  --- 59

  |       |       |        |  

61  --- |---- 67        |

  |       |       |        |

71  --  73 ----|---  79

          |       |        |

        83  ----| ---  89

                  |

                 97

.

One may calculate "distance" between two primes on this graph. Say, the distance between 11 and 23 is of 2 by optimal path; the distance between 43 and 97 is of 3 by a unique path. (There are many ways to study a graph, say, to define a matrix of connection). Note: there exists a concept of "prime graph" which is devised in the different sense (of finite groups).

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Problem. Estimate the number of primes owning at least one digit of degree one within [1, x] for sufficiently large x. (I call such primes "singular", denoted by Sp).

For x = 10^2, this number is of 1.

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So comes the experimental time. First of all, one needs to devise a computer code to calculate the degree of a targeted digit of a given prime. The basic idea is of replacement and factoration. As a first test, just take the 100th prime, 541. The degree of each digit of 541 can be illuatrated as follows.

.

           5 4 1

           |  |   |__ 547

           |  |____ 571 __ 521

           641

           |

           941

           |

           241

.

Apparently, 541 is not a singular prime. By a search from the table of primes with x = 10^4, one may find 483 singular primes out of 1229 primes. For x = 10^4, the largest sigular prime is 9973. An amazing feature of singular primes is that, indexed by the number of themselves, the plot of their values seemingly exhibits a nice linearity. That is to say, Sp(n) is of  quasi-linearity, where n is the index for the nth singular prime. This feature of quasi-linearity is observed from a linear fitting of the 483 singular primes within [1, 10^4]. (One can confirm the linearity by looking at the histogram of these 483 singular primes, shaped like a uniform distribution, which is apparently different from the histogram of the total 1229 primes within [1, 10^4]). More specifically, the number of Sp primes is estimated as #Sp ≈ (1+ o(1))·x /20. For x = 10^4, taking o(1) = 0 is ok; for x = 10^5 or 10^6, taking o(1) = 0.25 is ok. (A more close check is needed). The density of singular primes increases in the primes when k grows bigger (x = 10^k).

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Comment: the concept of "degree of digit" of primes is worthy of close studies.(For x = 10^6, the minimum gap between two consecutive singular primes is 2; eg., 419 and 421).

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

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