-indicator

-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}∪{∞}→Q_p , where Q_p 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 so-called "infinity prime", i.e. Q_∞, stands for the real number field R, and all negative numbers do not have square roots in R, we define:

_p(∞) := ∑_n∈N⁺ -pⁿ = p/(p-1)

Obviously, for any prime q (including "infinity prime"), the coefficients of the infinity sum above are periodic, thus the value of -indicator is definitely a rational number. Furthermore, _p(q) times (p^q-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).

See also

In other languages

• 本页中文版见-指示函数。