We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

We show that for any fixed ε > 0, there are numbers δ > 0 and p 0 2 with the following property: for every prime p p 0 , there is an integer p δ < N p 1/(4 √ e)+ε such that the sequence 1, 2,. .. , N contains at least δN quadratic non-residues modulo p. We then apply this result to obtain a new estimate on the smallest quadratic nonresidue in a Beatty sequence.

Plots of quadratic residues display some visual features that are analyzed mathematically in this paper. The graphical patterns of quadratic residues modulo m depend only on the residues of m modulo the lowest positive integers.

The projective special linear group PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n≡ 3 (mod 4) on the set {0,1, · · · ,n−1,∞}. It is known that the extended odd-like quadratic residue codes are invariant under PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n≡ 1 (mod 4), an infinite family of 3-designs for primes n≡ 3 (mod 4)...

The super-Poincaré covariant formalism for the superstring is used to compute mass-less four-point two-loop amplitudes in ten-dimensional superspace. The computations are much simpler than in the RNS formalism and include both external bosons and fermions.

(here b and a run over the quadratic nonresidues and quadratic residues, respectively, that lie between 0 and p) is a unit of the real quadratic field R(y/p), and that 77> 1. The fact that -n > 1 is usually deduced from the theory of the classnumber of quadratic fields. We present a short proof independent of the theory of the class-number. As in the paper of Chowla and Mordell [Note on the non...

Most of the poultry farms in Bangladesh use antibiotics to combat risk disease occurrences due poor managemental practices. This haphazard and illegitimate is responsible for antibiotic resistance residual problem eggs meat. However, data on consumption by layer birds farmers' perspectives are shockingly rare Bangladesh. In order understand perceptions usage Bangladesh's small-scale farms, this...

Miscanthus is an intriguing cellulosic bioenergy feedstock because its aboveground productivity is high for low amounts of agrochemical inputs, but soil temperatures below -3.5 °C could threaten successful cultivation in temperate regions. We used a combination of observed soil temperatures and the Agro-IBIS model to investigate how strategic residue management could reduce the risk of rhizome ...