[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。]

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

Why some papers are seemingly remembered forever, while others are forgotten?

♖      ♘   7          5

♔      ♗   2          3

Story - The princes appeared.

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

Recall: ...it is not clear that there is any simple property of individual digits of primes which is constrained in any way.

---- In the last note, "degree of digit" of a given prime is defined.

---- The concept was inspired by a casual problem to decide an unknown digit of an incomplete number (like 349?721) to acquire a prime.

---- "Singular primes" are those whose one of the digits is of one degree.

---- While such primes seemingly exhibit a quasi-linearity, their density in primes within [1, 10^k] increases with k.

---- The concern of rarity may draw one's attention to the "pure primes" whose digits are of 2, 3, 5 or 7.

---- Let the set of pure primes within [1, 10^k] be denoted by P0(k).

---- The discussion below takes P0(k) as the "world" of concern.

---- It is temping to search singular primes within P0(k), to gain a set of singular & pure primes, denoted by SP0(k).

---- That is to say, for the ith digit of a pure prime, one replaces the digit with (2), 3, (5), 7 to reach another pure prime, if any.

---- Note that, for the last digit, 2 and 5 are not applicable. 

---- To see a specific example, for k = 8 (i.e. x = 10^8), one has 6546 pure primes among the total 5,761,455 primes, taking a proportion of 0.11%.

---- (Note: for k = 7, pure primes occupy 0.29% of the total primes within [1, 10^7]).

---- Surprisingly, the singular subset SP0(8) of P0(8) contains 6535 pure primes, taking a proportion of 99.83% of P0(8) !

---- To pursue rarity, a natural consideration is to turn to search pure primes whose each digit is of one degree.

---- There are 366 such primes within P0(8).

---- One may call such primes as "super singular pure primes", denoted by SSP0(k).

---- So, #SSP0(8) = 366, taking a proportion of 5.59% of P0(8).

---- By the concept of degree of digit, as shown in the last note, primes of equal digit size generally form a connected graph (in the sense of digit altering).

---- Now, each member in SSP0(k) forms an "island", disconnected to any other member in P0(k) by digit altering.

---- One may give a nick name of "prince" for the members in SSP0(k).

---- The first prince is 37 while the last prince is 77757257, within P0(8).

---- The minimum gap between two consecutive princes is of 6, while the largest gap of two consecutive princes is 14470230.

---- The most frequently occurred gap is 24 (3 occurrences).

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Comment: while "princes" are isolated from any member in P0(k), it is possible to find some usual primes (like "friends") connected to them...

♙ 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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Terms of awareness/ use