查看“︁-indicator”︁的源代码

'''-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.

For any odd prime ''p'', define the -indicator <sub>''p''</sub> : {''q'':''q'' is a prime}∪{∞}→'''Q'''<sub>''p''</sub>, where '''Q'''<sub>''p''</sub> is the ''p''-adic number field:

<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:

<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).

Here are some examples:
* <sub>3</sub>(2)=-12/41
* <sub>3</sub>(3)=-3/13
* <sub>3</sub>(5)=-24/121
* <sub>3</sub>(7)=-453/1093
* <sub>3</sub>(11)=-24249/88573
...

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).