[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.

The art of a theorist is not to solve problems, but convince those who solve problems.

♖    ♘   7        5

♔    ♗   2        3

Story - The friends of princes are all civilians, yet highly educated.

♙ 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.

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                      Singular primes

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---- The degree of the ith digit of a given prime is the number of primes acquired by altering that digit*.

---- Let Ai,s(p) denote the operation of altering the ith digit of the prime p to the digital number s, so that Ai,s(p) is also a prime.

---- For example, A1,1(23) = 13,  A1,4(23) = 43, A1,5(23) = 53,  A1,7(23) = 73, A1,8(23) = 83.

---- In particular, A1,2(23) = 23.

---- The digital number s is called a "gate" of the prime p at the ith digit, if Ai,s(p) is a prime other than p.

---- Apparently, the degree of the ith digit of the prime p is the number of primes available when other digits are freezed.

---- Let di(p) denote the degree of the ith digit of the prime p.

---- One has d1(23) = 6, with 5 gates at the first digit.

---- A prime is called "singular" if the prime has at least one digit of degree one.

---- One may check A2,9(23) = 29, A2,3(23) = 23.

---- That is, one has d2(23) = 2, with 1 gate at the second digit.

---- Therefore, 23 is not a singular prime.

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        2 3

                              1,4,5,7,8     9

The prime 23 and its digital gates to enter related primes.

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---- One may check 97, rare within [1, 100], is a singular prime, as d2(97) = 1. 

---- That is, at the second digit, one cannot find a digital number s other than 7, such that A2,s(97) is a prime other than 97.

---- The prime 97 does not have a gate at the second digit, forming a singular prime.

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                      Super singular pure primes ("princes")

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Surprisingly, singular primes are not rare among the usual primes within [1, x], where x is sufficiently large.

---- For x = 10^8, there are totally 5,501,550 singular primes among the 5,761,455 usual primes within [1, x], taking a proportion of ~ 95.5% ! .

---- To pursue rarity, one may consider singular primes in the realm of pure primes whose digits are of 2, 3, 5 or 7.

---- There are only 6546 pure primes within [1, 10^8].

---- Surprisingly, there are 6535 singular primes among these pure primes, taking a proportion of ~ 99.8%.

---- The expected rarity does not exist.

---- So comes the concept of "super singular primes" whose each digit is of degree one.

---- There are 366 super singular primes among the pure primes within [1, 10^8], taking a proporation of 5.13% ——

---- (That is to say, ~95% pure primes are connected by the operation of digit altering).

---- The first ten of them are 37,  337,  23333,  25523,  53233,  222773,  232753,  252277,  253537, and 255757.

---- The last ten of them are 77352337,  77353327,  77355533,  77375257,  77525737,  77555227,  77572577,  77737733,  77753573,  and 77757257.

---- Such primes are nicknamed as "princes", denoted by SSP0(k), with x = 10^k.

---- As each digit is of degree one for any prime in SSP0(k), such primes have no gate at any digit to enter other pure primes.

---- In the view of digit altering, princes are not only isolated from each other, but isolated from any other pure primes.

---- However, it is expectable for princes to find gates at some digits to enter the non-pure primes (of "civilian" nature).

---- For example, prince 37, one has ——

---- A1,1(37) = 17,  A1,4(37) = 47,  A1,6(37) = 67, A1,9(37) = 97.

---- These primes are "connected" to prince 37 by the gates 1, 4, 6 or 9 at the first digit.

---- Like wise, A2,1(37) = 31.

---- That is, the prime 31 is connected to prince 37 by the gate 1 at the second digit.

---- One may give these 5 primes the nickname of "friends" of prince 37, reached by digit altering outside the realm of pure primes (of "royal" nature).

---- In this sense, the friends of princes are all civilians.

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                                      1,4,6,9     1

The price 37 and its digital gates to reach his civilian friends.

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Finally, one may step back from the pure primes and search super singular primes among usual primes within [1, x]. For x = 10^6, there are totally 6 of them, i.e. 294001,  505447,  584141,  604171,  929573,  and 971767. None of them is a pure prime. Such primes are absolutely isolated from each other and all other primes with respect to the operation of digit altering. These primes might be used as "gold" in the kingdom of primes.

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Comments: There are a number of studies might be considered surrounding the concepts of "degree of digit" and "pure primes". For example, one may define the "degree of prime" by the sum of degrees of digits of a prime. Preliminary studies indicate that the degrees of pure primes can be well fitted by the t Location-Scale distribution. Other related concepts can be devised and studied.

♙ 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