Input: An integer n > 1. 0: if n is a power then output composite fi; 1: r := 2; 2: while (r < n) do 3: if gcd(r, n) 6= 1 then output composite fi; 4: if r is prime then 5: q := largest prime factor of r − 1; 6: if (q ≥ 4√r log n) and (n(r−1)/q 6≡ 1 mod r) then break fi; 7: fi; 8: r := r + 1; 9: od; 10: for a = 1 to 2 √ r log n do 11: if (x− a)n 6≡ (xn − a) mod (xr − 1, n) then output composite...

let $cr_{n}(f)$ denote the algebra of $n times n$ circulant matrices over the field $f$. in this paper, we study the unit group of $cr_{n}(f_{p^m})$, where $f_{p^m}$ denotes the galois field of order $p^{m}$, $p$ prime.

For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with 2$ and $m\geq $k\in [0,n-1]$. It is known that any sequence $mn+n-1+k$ terms from $G$ must contain nontrivial zero-sum length at most $mn+n-1-k$. The associated inverse question to characterize those sequences maximal $mn+n-2+k$ fail subsequence $k\leq this the for Davenport Constant. $k=n-1$, $\eta(G)$ i...

Chronic Constipation is a debilitating condition that is often associated with different abdominal problem. It can cause distress for the child and family and can result in emotional disturbance and family problem. Based on the current algorhytm, the treatment of chronic constipation consists of 4 important phases, 1: education, 2: disimpaction, 3: prevention of re-accumulation of feces and 4: ...

Let G=Cn×Cn, where Cn denotes a cyclic group of order n, and let k∈[0,n−1]. We study the structure sequences terms from G with maximal length |S|=2n−2+k that fail to contain nontrivial zero-sum subsequence at most 2n−1−k. For k≤1, this is inverse question for Davenport Constant. k=n−1, η(G) invariant concerning short subsequences. The in both these cases (known respectively as Property B C) was...

Let n denote a positive integer relatively prime to 10. Let the period of 1/n be a · b with b > 1. Break the repeating block of a · b digits up into b sub blocks, each of length a, and let B(n, a, b) denote the sum of these b blocks. In 1836, E. Midy proved that if p is a prime greater than 5, and the period of 1/p is 2 · a, then B(p, a, 2) = 10 − 1. In 2004, B. Ginsberg [2] showed that if p is...

If n− 1 is prime, then the lowest-terms numerator is clearly either 1 or the prime n−1 (and in fact the latter, since n−1 is relatively prime to n and to (n−2)!). If n− 1 is composite, either it can be written as ab for some a 6= b, in which case both a and b appear separately in (n− 2)! and so the numerator is 1, or n− 1 = p2 for some prime p, in which case p appears in (n− 2)! and so the nume...

In this article we give the definition of weakly prime hyperideals over multiplicative hyperring. We provide important results showing relations between and hyperideals. Then n-hyperideal (1,n) hyperideal hyperrings. investigate their properties some examples. Hyperstructures are very topic because it satisfy to study multidisiplinary. Especially in chemistry, coding theory, geometry, artificia...

let $r$ be a ring and $m$ a right $r$-module with $s=end_r(m)$. a module $m$ is called semi-projective if for any epimorphism $f:mrightarrow n$, where $n$ is a submodule of $m$, and for any homomorphism $g: mrightarrow n$, there exists $h:mrightarrow m$ such that $fh=g$. in this paper, we study sgq-projective and$pi$-semi-projective modules as two generalizations of semi-projective modules. a m...