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