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[March for reflection |Maynard] evasion from reality

已有 416 次阅读 2022-7-19 22:48 |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.

If digits are made of iron, why primes are free from magnet?

    ♘   7        5

 

    ♗   2        3

Story - a plain day past.

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

4. Restricted digits of primes

Many basic and fundamental properties of primes can be recast as problems about their digits.

---- properties of primes ~ problems about digits

---- Question: Will digits offer more subtle description to primes? Will digits offer a finer "resolution" or more information to primes?

---- Problem: Given a number of k digits, with one of the digit unknown. Is there a number of 0 ~ 9 such that the given number to be a prime?

---- Say, given the number 349x721, one may try to factorate the possible 10 numbers... The answer is positive for this case —— 3499721.

---- Try another case: 3x93721. One can find x = 3, 6, 9 is the answer.

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The problem of how many primes there are in a short interval [x, x + x^θ] is essentially equivalent to asking whether there are primes with ⌈logx / log10⌉ digits which have a specified string of ⌈θ·logx / log10⌉ numbers as their leading decimal digits.

---- Consider x = 33, θ = 2. The interval is of [33, 33+1089] which contains 187 - 11 = 176 primes, such as 37, 41, 43, 47, ...,1103, 1109, 1117.

---- One calculates ⌈logx / log10⌉ = 2, ⌈θ·logx / log10⌉ = 4.

---- Whether there are primes with 2 digits which have a specified string of 4 numbers as their leading decimal digits ——

---- "primes with 2 digits" are 11    13    17    19    23    29    31    37    41    43    47    53    59    61    67    71    73    79   83    89    97.

---- "a specified string of 4 numbers" ?? What does this mean ??

.

Comment: lack of specific examples leads to artificial puzzles...

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Similarly, asking for primes with k decimal digits which end in a specific string of θk digits is asking for primes less than 10^k in an arithmetic progression modulo 10^θk.

---- If one takes k = 3, θ = 2, "asking for primes with k decimal digits" refers to primes like xxx.

---- "a specific string of θk digits" refers to a string like ssssss.

---- primes of form xxx end in a string of form ssssss?

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The question as to whether there are infinitely Mersenne primes is equivalent to asking whether there are infinitely many primes with no digit 0 in their base 2 expansion.

---- Mersenne primes are of the form 2^p - 1.

---- Say, take p = 2, one gains the first Mersenne prime 3.

---- Take p = 3, one gains Mp = 7.

---- Take p = 5, one gains Mp = 31.

---- In base 2, the number 31 takes the form of 11111.

(This sentence is clear and apparent).

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The final decimal digit of a prime number larger than 10 must be 1, 3, 7 or 9, since any integer ending in 0, 2, 4, 5, 6 or 8 must be a multiple of 2 or 5.

--- clear.

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More generally, the final digit in base b of a prime p > b must be coprime to b.

---- Set b = 2. 

---- Examples of primes greater b:  3, 5, 7 (decimal form), or 11, 101, 111 (binary form). 

.

Beyond this condition, however, it is not clear that there is any simple property of individual digits of primes which is constrained in any way.

---- Maynard neglected digit 0 (which never presents as a leading digit in any prime).

---- Just a minor leakage.

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We might guess that for any large set A ⊆ [X, 2X] defined only in terms of base b digital properties and containing only integers with last digit coprime to b, the density of primes in A is the same as the density of primes in the set of all integers in [X, 2X] which have last digit coprime to b.

---- A ⊆ {n : restricted digital properties } ∩ [n: last digit coprime to b] ∩ [X, 2X].

---- Denote the set [n: last digit coprime to b] ∩ [X, 2X] as X.

---- Density of p(A) is the same as the density of p(X).

---- Maynard means that the prescribed conditions on the digital level for numbers might not affect the proportion of primes. 

---- This is a point that could be verified by computer experiments.

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This density is ::: b/(φ(b)logX), so we might guess that #{p ∈ A} ≈ b/(φ(b)logX)·#A in this case.

---- Another chance for experiments.

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Theorems 2.2 and 2.3 confirm this heuristic for the set A of integers with some prescribed binary digits, or the set of integers with a digit missing in their decimal expansion.

---- clear.

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(Note that #A ≈ X^log(b-1)/logb for numbers with no base b digit equal to a0.)

---- To be tested.

 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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