-indicator：修订间差异
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'''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues. | '''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping [https://mathworld.wolfram.com/PrimeNumber.html prime numbers] to [https://mathworld.wolfram.com/p-adicNumber.html ''p''-adic numbers], indicating how "similar" two primes are, in the sense of [https://mathworld.wolfram.com/QuadraticResidue.html quadratic residues]. | ||
==Definition== | ==Definition== | ||
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<sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup> | <sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup> | ||
where Kronecker means the Kronecker symbol; for ''q''=∞, since the local field on the "infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers are not square roots in '''R''', we define: | where Kronecker means the [https://mathworld.wolfram.com/KroneckerSymbol.html Kronecker symbol]; for ''q''=∞, since the local field on the "infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers are not square roots in '''R''', we define: | ||
<sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1) | <sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1) | ||
Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a rational number. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1). | Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a [https://mathworld.wolfram.com/RationalNumber.html rational number]. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1). | ||
==Examples== | ==Examples== | ||
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==Usage== | ==Usage== | ||
For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square). | For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the [https://mathworld.wolfram.com/p-adicNorm.html ''p''-adic distance] |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square). | ||
2025年3月15日 (六) 11:04的版本
-indicator (pronunciation: Eye-indicator, Shidinn language: - ) is a mathematical function mapping prime numbers to p-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.
Definition
For any odd prime p, define the -indicator p : {q:q is a prime}∪{∞}→Qp , where Qp is the p-adic number field:
p(q) := ∑n∈N+ Kronecker(-n|q) pn
where Kronecker means the Kronecker symbol; for q=∞, since the local field on the "infinity prime", i.e. Q∞, stands for the real number field R, and all negative numbers are not square roots in R, we define:
p(∞) := ∑n∈N+ -pn = p/(p-1)
Obviously, for any prime q (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a rational number. Furthermore, p(q) times (pq-1) is an integer if q is a finite odd prime, and so is p(2) times (p8-1).
Examples
Here are some examples for p=3:
- 3(2)=-12/41
- 3(3)=-3/13
- 3(5)=-24/121
- 3(7)=-453/1093
- 3(11)=-24249/88573
- ...
Usage
For any two "primes" q1 and q2 (no matter finite or infinite), the p-adic distance |p(q1)-p(q2)|p measures how similar q1 and q2 are. If |p(q1)-p(q2)|p < p-n for a positive integer n, then we can obtain that both negative integers -1..-n and positive integers 1..n (think why?) "performs" consistently on q1-adic and q2-adic (about whether the integer is a square).