In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. Definition The von Mangoldt function, conventionally written as Λ(n), is defined as :\Lambda(n) = \begin{cases} \log p & \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 & \mbox{otherwise.} \end{cases} which is a sequence starting: :\log 1 , \log 2 , \log 3 , \log 2 , \log 5 , \log 1 , \log 7 , \log 2 , \log 3,... [http://oeis.org/A014963 oeis sequence A014963] It is an example of an important arithmetic function that is neither multiplicat...

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the reciprocity law: Let p, q > 2 be two distinct prime numbers. Then (Supplement 1) :x2 ≡ −1 (mod p) is solvable if and only if p ≡ 1 (mod 4). (Supplement 2) :x2 ≡ 2 (mod p) is solvable if and only if p ≡ ±1 (mod 8). (Quadratic reciprocity) Let q* = ±q   where the sign is plus ...