Abstract Let N be a $$\mathbb {Z}$$ Z -nearalgebra; that is, left nearring with identity satisfying $$ k(nn^{\prime })=(kn)n^{\prime }=n(kn^{\prime })$$ k ( n ′ ) =</mml:m...

n = 1: 1 = 0 + 1; n = 2 (prime): 2 = 1 + 1; n = 3 (prime) is not a sum of two squares. n = 4: 4 = 2 + 0. n = 5 (prime): 5 = 2 + 1. n = 6 is not a sum of two squares. n = 7 (prime) is not a sum of two squares. n = 8: 8 = 2 + 2. n = 9: 9 = 3 + 0. n = 10: 10 = 3 + 1. n = 11 (prime) is not a sum of two squares. n = 12 is not a sum of two squares. n = 13 (prime): 13 = 3 + 2. n = 14 is not a sum of t...

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [ {;} be a function, where S(M) is the set of submodules ofM. Suppose n 2 is a positive integer. A proper submodule P of M is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 R and x 2 M and a1 . . . an−1x 2P(P), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x 2 P...

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote Rogers-Ramanujan fraction having well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$ If \begin{align*} \sum_{n=0...

n>0 (1− q) be Dedekind’s eta-function. For a prime p, denote by U Atkin’s Up-operator. We say that a function φ with a Fourier expansion φ = ∑ u(n)q is congruent to zero modulo a power of a prime p, φ = ∑ u(n)q ≡ 0 mod p if all its Fourier expansion coefficients are divisible by this power of the prime; u(n) ≡ 0 mod p for all n. In this paper we prove the following congruences. Theorem 1. (i) I...

Yes, but such a formula is complicated. For example, is there a polynomial f ∈ Z[x] for which f(n) = pn? f(x) = anx n + · · ·+ a1x+ a0 f(a0) = ana0 n + · · ·+ a1a0 + a0 so a0 | f . Suppose q is prime and f(n) = q. Then q | f(n + kq) for each k ∈ Z. So, in particular, we see that if f(m) is prime for each positive integer m, then f is a constant. In particular, f(x) = q for some prime q. The pol...

Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...

Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...