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We show that the quartic Diophantine equations $ax^4+by^4=cz^2$ has only trivial solution in the Gaussian integers for some particular choices of $a,b$ and $c$. Our strategy is by elliptic curves method. In fact, we exhibit two null-rank corresponding families of elliptic curves over Gaussian field. We also determine the torsion groups of both families.

In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $$a^2+b^4$$ a 2 + b 4 . To show this they used the Jacobi symbol to define the spin of Gaussian integers, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $${\mathbb {Z}}[\zeta _{12}]={\mathbb {Z}}[\zeta _3,i]$$ Z [ ζ 12 ] = Z [ ζ 3 , i ] by using the cubic residue character on the Eisenstein integers $${\mathbb {Z}}[\zeta _3]$$ Z [ ζ 3 ] . Our main theorem says that the cubic spin is equidistributed along prime ideals of $${\mathbb {Z}}[\zeta _{12}]$$ Z [ ζ 12 ] . The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $$a^2+b^6$$ a 2 + b 6 on the Eisenstein integers.

Let $\mathrm{g}_1 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of square-free factors. Let $\mathrm{g}_2 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of power-free factors. In this paper we consider their summatory functions $\sum_{N(\alpha)\le x} \mathrm{g}_1 (\alpha)$ and $\sum_{N(\alpha)\le x} \mathrm{g}_2 (\alpha)$ and obtain asymptotic formulas for them. Also, we prove analogue of K\'{a}tai-Subbarao theorem to study the distribution of $\mathrm{g}_2 (\alpha)$ in increasing norm order case.

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

Recently, there is an increased interest in the study of modulo analog to digital converters (ADCs). These new systems can reconstruct a signal whose amplitude is much higher than the conventional ADC's dynamic range. Modulo ADCs are characterized by their modulo threshold and in the current literature, all existing works are limited to real-valued moduli. In this paper, we propose multi-channel modulo samplers with complex-valued moduli to sample a band-limited complex signal. Specifically, we discuss the construction of complex divisors from Gaussian integers and propose their efficient implementations. A memory-efficient, closed-form recovery algorithm is also proposed. Simulation results demonstrate that the proposed systems can provide stable reconstruction of a high dynamic range complex-valued signal at low sampling rates.

This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials., Comment: 21 pages

The Compute-and-Forward protocol in quasi-static channels normally employs lattice codes based on the rational integers $\mathbb{Z}$, Gaussian integers $\mathbb{Z}\left[i\right]$ or Eisenstein integers $\mathbb{Z}\left[\omega\right]$, while its extension to more general channels often assumes channel state information at transmitters (CSIT). In this paper, we propose a novel scheme for Compute-and-Forward in block-fading channels without CSIT, which is referred to as Ring Compute-and-Forward because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Thanks to the multiplicative closure of the algebraic lattices employed, a relay is able to decode an algebraic-integer linear combination of lattice codewords. We analyze its achievable computation rates and show it outperforms conventional Compute-and-Forward based on $\mathbb{Z}$-lattices. By investigating the effect of Diophantine approximation by algebraic conjugates, we prove that the degrees-of-freedom (DoF) of the optimized computation rate is ${n}/{L}$, where $n$ is the number of blocks and $L$ is the number of users., Comment: IEEE Transactions on Information Theory, to appear

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + \alpha D$ for $\alpha \in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.

Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$. To any of the 8 possible representations (a,b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950's Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance., Comment: Fixed a problem in Lemma 6.11, pointed out by Corentin Perret-Gentil

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2 . Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.

In this paper, we examine Gaussian neighborhood-prime labeling of generalized Peterson graph and graphs which contain Hamiltonian cycle.

The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.

We solve Cauchy problems for discrete holomorphic functions defined on the Gaussian integers, which leads to the existence of discrete holomorphic functions with arbitrarily fast growth. This proves that certain classes of functions are closed in the sense of mathematical morphology.

A mixed graph $G$ is a graph obtained from a simple undirected graph by orientating a subset of edges. $G$ is self-converse if it is isomorphic to the graph obtained from $G$ by reversing each directed edge. For two mixed graphs $G$ and $H$ with Hermitian adjacency matrices $A(G)$ and $A(H)$, we say $G$ is $\mathbb{R}$\emph{-cospectral} to $H$ if, for any $y\in \mathbb{R}$, $yJ-A(G)$ and $yJ-A(H)$ have the same spectrum, where $J$ is the all-one matrix. A self-converse mixed graph $G$ is said to be determined by its generalized spectrum, if any self-converse mixed graph that is $R$-cospectral with $G$ is isomorphic to $G$. Let $G$ be a self-converse mixed graph of order $n$ such that $2^{-\lfloor n/2\rfloor}\det W$ (which is always a real or pure imaginary Gaussian integer) is square-free in $\mathbb{Z}[i]$, where $W=[e,Ae,\ldots,A^{n-1}e]$, $A=A(G)$ and $e$ is the all-one vector. We prove that, for any self-converse mixed graph $H$ that is $\mathbb{R}$-cospectral to $G$, there exists a Gaussian rational unitary matrix $U$ such that $Ue=e$, $U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. In particular, if $G$ is an ordinary graph (viewed as a mixed graph) satisfying the above condition, then any self-converse mixed graph $H$ that is $\mathbb{R}$-cospectral to $G$ is $G$ itself (in the sense of isomorphism). This strengthens a recent result of the first author., 22 pages,1 figure

Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Mart\'{i}nez [IEEE Trans. Inform. Theory, {\bf 62} (2016), 1183--1192], showed that for every prime number $p>5$ such that $p\equiv \pm 5 \pmod{12}$, the Cayley graph $\mathcal{G}_p=\textnormal{Cay}(\Z_p[i], S_2)$, where $S_2$ is the set of units of $\Z_p[i]$, induces a 2-quasi-perfect Lee code over $\Z_p^m$, where $m=2\lfloor \frac{p}{4}\rfloor$. They also conjectured that $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$. In this paper, we solve this conjecture. Our main tools are Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lov\'{a}sz from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the famous Golomb--Welch conjecture on the existence of perfect Lee codes.

The security of RSA relies on the computationally challenging factorization of RSA modulus with being a large semi-prime consisting of two primes for the generation of RSA keys in commonly adopted cryptosystems. The property of both congruent to 1 mod 4, is used in Euler&rsquo, s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares . As becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research.

We generalize a Theorem of Ricci and count Gaussian primes $\mathfrak{p}$ with short interval restrictions on both the norm and the argument of $\mathfrak{p}$., 18 pages

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph $\mathcal{F}$ , whose vertices are reduced Gaussian rationals in $\mathbb{Q}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in $\mathbb{H}^{3}$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of $\mathbb{H}^{3}$ by Farey octahedrons. A natural distance $\varrho $ on $\mathbb{Q}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty $ to $w\in \mathbb{Q}(i)$ , where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$ . The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$ , $n$ , and $n+1$ . Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any $w$ is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog–Biro–Cherubini–Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

Recently, Bennis and others studied an extension of the zero-divisor graph of a commutative ring R. They called this extension the extended zero-divisor graph of R, denoted by Γ(R). The graph Γ(R) has as set of vertices all the nonzero zero-divisors of R, Z(R)*, and two distinct vertices x and y are adjacent if there are nonnegative integers n and m such that xnym = 0 with xn ≠ 0 and ym ≠ 0. In this paper, we study several properties of the extended zero-divisor graph of the ring of Gaussian integers modulo n (Γ(Zn[i])). We characterize the positive integers n such that Γ(Zn[i]) = Γ(Zn[i]). The diameter and girth, as well as the positive integers n such that Γ(Zn[i]) is planar or outerplanar, are also determined.

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenvalues of $\ell$-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties., Comment: 41 pages. Taking into account the referees' comments. To appear in Algebra & Number Theory

The pslq algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend pslq to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). We outline the algorithm, discuss the required modifications for handling algebraic integers, problems that have arisen, experimental results, and challenges to further work.

In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

We study a problem that arises naturally in the discrete quantum computation model introduced in Gatti and Lacalle (Quantum Inf Process 17:192, 2018). Given an orthonormal system of discrete quantum states of level k $$(k\in \mathbb {N})$$, can this system be extended to an orthonormal basis of discrete quantum states of the same level? This question turns out to be a difficult problem in number theory with very deep implications. In this article, we focus on the simplest version of the problem, 2-qubit systems with integers (instead of Gaussian integers) as coordinates, but with normalization factor $$\sqrt{p}$$ $$(p\in \mathbb {N}^*)$$, instead of $$\sqrt{2^k}$$, being p a prime number. With these simplifications, we prove the following orthogonal version of Lagrange’s four-square theorem: Given a prime number p and $$v_1,\dots ,v_k\in \mathbb {Z}^4$$, $$1\le k\le 3$$, such that $$\Vert v_i\Vert ^2=p$$ for all $$1\le i\le k$$ and $$\langle v_i|v_j\rangle =0$$ for all $$1\le i

In this paper, we apply the continued fraction method to launch an attack on the three RSA-type cryptosystems when the private exponent d is sufficiently small. The first cryptosystem, proposed by Kuwakado, Koyama and Tsuruoka in 1995, is a scheme based on singular cubic curves y 2 = x 3 + b x 2 ( mod N ) where N = p q is an RSA modulus. The second cryptosystem, proposed by Elkamchouchi, Elshenawy and Shaban in 2002, is an extension of the RSA scheme to the field of Gaussian integers using a modulus N = P Q where P and Q are Gaussian primes such that p = | P | and q = | Q | are ordinary primes. The third cryptosystem, proposed by Castagnos in 2007, is a scheme over quadratic field quotients with an RSA modulus N = p q based on Lucas sequences. In the three cryptosystems, the public exponent e is an integer satisfying the key equation e d − k ( p 2 − 1 ) ( q 2 − 1 ) = 1 . Our attack is applicable to primes p and q of arbitrary sizes and we do not require the usual assumption that p and q have the same bit size. Thus, this is an extension of our recent result presented at ACISP 2016 conference. Our experiments demonstrate that for a 513-bit prime p and 511-bit prime q, our method works for values of d of up to 520 bits.

Based on p-ary pseudorandom sequences, this study proposes a construction of degree-k Gaussian integer sequences of period N = k ( p m − 1 ) / ( p − 1 ) by utilising kth power residue symbol satisfying k | ( p − 1 ) , where p is an odd prime and positive integers m , k ≥ 2 . The periodic autocorrelation values are 0 at shifts τ ≢ 0 ( mod ( N / k ) ) of the resultant sequences. Specially, there is exactly one non-zero out-of-phase periodic autocorrelation value of the resultant sequences for k = 2 . The non-zero elements of the sequences are balanced and can be predefined flexibly. Moreover, the maximum energy efficiency of the proposed sequences is close to ( p − 1 ) / p for sufficiently large m.

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^{3}-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x^{2}-y^{2}=n^{2} for some Gaussian integer n. The analog of this statement over Z remains open., arXiv admin note: text overlap with arXiv:1303.4329 by other authors

In this paper, we aim to give full or partial proofs for the following three conjectures of V. J. W. Guo and C. Krattenthaler: (1) Let a > b be positive integers, α , β be any integers and p be a prime satisfying gcd ⁡ ( p , a ) = 1 . Then there exist infinitely many positive integers n for which ( a n + α b n + β ) ≡ r ( mod p ) for all integers r ; (2) For any odd prime p , there are no positive integers a > b such that ( a n b n ) ≡ 0 ( mod p n − 1 ) for all n ≥ 1 ; (3) For any positive integer m , there exist positive integers a and b such that a m > b and ( a m n b n ) ≡ 0 ( mod a n − 1 ) for all n ≥ 1 .

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of $N = qp$ , where $q$ and $p$ are distinct primes. The study begins with the partitioning of a ring $\mathbb {Z}_{N}$ into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of $\mathbb {Z}_{N}$ should be carried out; here, two cases, the even period $N = 2p$ and the odd period $N = qp$ , are treated separately. We can adopt the Legendre sequences of the prime period $p$ to construct PGISs of period $2p$ with degrees larger than four. For the case of period $qp$ , we introduce the Jacobi symbols to partition $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.

Let N = p q be an RSA modulus with unknown factorization. Some variants of the RSA cryptosystem, such as LUC, RSA with Gaussian primes and RSA type schemes based on singular elliptic curves use a public key e and a private key d satisfying an equation of the form e d − k ( p 2 − 1 ) ( q 2 − 1 ) = 1 . In this paper, we consider the general equation e x − ( p 2 − 1 ) ( q 2 − 1 ) y = z and present a new attack that finds the prime factors p and q in the case that x, y and z satisfy a specific condition. The attack combines the continued fraction algorithm and Coppersmith's technique and can be seen as a generalization of the attacks of Wiener and Blomer–May on RSA.

The code-based McEliece and Niederreiter cryptosystems are promising candidates for post-quantum public-key encryption. Recently, q-ary concatenated codes over Gaussian integers were proposed for the McEliece cryptosystem, together with the one-Mannheim error channel, where the error values are limited to the Mannheim weight one. Due to the limited error values, the codes over Gaussian integers achieve a higher error correction capability than maximum distance separable (MDS) codes with bounded minimum distance decoding. This higher error correction capability improves the work factor regarding decoding attacks based on information-set decoding. The codes also enable a low complexity decoding algorithm for decoding beyond the guaranteed error correction capability. In this work, we extend this coding scheme to codes over Eisenstein integers. These codes have advantages for the Niederreiter system. Additionally, we propose an improved code construction based on generalized concatenated codes. These codes extend to the rate region, where the work factor is beneficial compared to MDS codes. Moreover, generalized concatenated codes are more robust against structural attacks than ordinary concatenated codes.

This paper presents the construction of perfect Gaussian integer sequence (PGIS) of period $N=p^{k}$ with degrees equal to or less than $k+1$ , where $p$ is a prime number and $k\geq 2$ . The study begins with the partitioning of $\mathbb {Z}_{N}$ into $k+1$ subsets, from which $k+1$ base sequences are defined to construct degree- $(k+1)$ PGISs. The $k$ constraint equations derived from matching the ideal periodic autocorrelation function criterion in the time domain are nonlinear. We propose a decomposition method that can transform these $k$ nonlinear equations into $2k-2$ linear equations with $2k-2$ variables. The number of nonlinear constraint equations derived from the frequency domain is $k+1$ , where the $k+1$ sequence coefficients can be derived sequentially by an iteration algorithm. These two proposed schemes simplify the construction of the degree- $(k+1)$ PGIS. PGISs having less than $k+1$ degrees can be constructed in two steps. First, some sequence coefficients are assigned a zero value or one or more pairs of two consecutive base sequences are combined into one. The second step involves the adjustment of new sequence coefficients for the associated sequence to be perfect. We show that there exist at least $2k+(({k(k-1)(k-2)})/{6})$ distinct PGIS patterns of period $N=p^{k}$ with degrees equal to or less than $k+1$ .

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite quadratic form represents all positive integers coprime to $3$ and $\leq 290$, then it represents all positive integers coprime to $3$. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between $1$ and $451$ represents all positive odd integers., Comment: 15 pages

We show that there are infinitely many distinct rational primes of the form p 1 = a 2 + b 2 and p 2 = a 2 + ( b + h ) 2 , with a , b , h integers, such that | h | ≤ 246 . We do this by viewing a Gaussian prime c + d i as a lattice point ( c , d ) in R 2 and showing that there are infinitely many pairs of distinct Gaussian primes ( c 1 , d 1 ) and ( c 2 , d 2 ) such that the Euclidean distance between them is bounded by 246. Our method, motivated by the work of Maynard [9] and the Polymath project [13] , is applicable to the wider setting of imaginary quadratic fields with class number 1 and yields better results than those previously obtained for gaps between primes in the corresponding number rings.

Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a square torus that we conjecture to be the most dense for certain numbers of packing disks, using continued fractions to approximate $1/\sqrt{3}$ and $2-\sqrt{3}$. We also define a constant to measure the efficiency of a packing motived by a related constant due to Markov for continued fractions. One idea is to use the unique factorization property of Gaussian integers to prove that there is an upper bound for the Markov constant for grid-like packings. By way of contrast, we show that an upper bound by Peter Gruber for the error for the limiting density of a packing of equal disks in a planar square, which is on the order of $1/\sqrt{N}$, is the best possible, whereas for our examples for the square torus, the error for the limiting density is on the order of $1/N$, where $N$ is the number of packing disks., Comment: Various small mistakes were corrected, and the reference to Gruber's result was more accurately stated