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...

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

The discrepancies between the measurements of rare (semi-)leptonic $B$ decays and corresponding Standard Model predictions point convincingly towards existence new physics for which a heavy neutral gauge boson ($Z^\prime$) is prime candidate. However, effect mixing $Z^\prime$ with SM $Z$, even though it cannot be avoided by any symmetry, usually assumed to small thus neglected in phenomenologic...

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian. In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, i?-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asympt...

We present the rst eecient statistical zero-knowledge protocols to prove statements such as: { A committed number is a prime. { A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p ? 1)=2 and (q ? 1)=2 are prime. { A given integer has large multiplicative order modulo a composite number that consists of two safe prime factors. The main building b...

This paper presents the first efficient statistical zero-knowledge protocols to prove statements such as: A committed number is a pseudo-prime. A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p1)=2 and (q 1)=2 are primes as well. A given value is of large order modulo a composite number that consists of two safe prime factors. So far, nomethod...

Let π be a cusp form on GL(n)/Q , i.e., a cuspidal automophic representation of GL(n,A ), where A denotes the adele ring of Q . We will say that a prime p is a Ramanujan prime for π iff the corresponding πp is tempered. The local component πp will necessarily be unramified for almost all p, determined by an unordered n-tuple {α1,p, α2,p, . . . , αn,p} of non-zero complex numbers, often represen...

One denotes by S(d) the collection of all torsion primes of degree d. The well known strong uniform boundedness conjecture states that for every d, the set S(d) is nite. The main results of [KaMa-92] summarize as follows: for all d, S(d) is of density zero, and for d 8, S(d) is nite. It should be mentioned that the original conjecture is not restricted to prime levels, but in [KaMa-92] it is sh...