Square Of Prime Research Articles

Flag-transitive point-primitive symmetric 2-designs of prime square order

Flag-transitive point-primitive symmetric 2-designs of prime square order

On the Moments of the Number of Representations as Sums of Two Prime Squares

Abstract We study the moments of the function that counts the number of representations of an integer as sums of two prime squares. We refine some of the previous arguments and apply the Selberg sieve to get an unconditional upper bound for all moments. We also prove a lower bound for all moments conditional on some generalization of the Green-Tao theorem on linear equations in primes. More precisely, for the fifth moment and onward, we get the expected order of magnitude lower and upper bounds. In addition, we provide some heuristics on the mass function of this representation function.

Prime square order Cayley graph of cyclic groups

In this paper, we consider the prime square order Cayley graph of a cyclic group [Formula: see text] of order [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct prime numbers. We discuss the Eulerianity and Hamiltonicity of the graph. Moreover, we present the girth, the diameter, the independence number, the chromatic number, the clique number, and the dominating number of the graph.

Additive problems with almost prime squares

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

A pair of equations in two prime squares, four prime cubes and powers of two

A pair of equations in two prime squares, four prime cubes and powers of two

Two prime squares, four prime cubes and powers of 2

Two prime squares, four prime cubes and powers of 2

Vinogradov’s theorem with Fouvry–Iwaniec primes

We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas.

Flag-transitive 2-designs with prime square replication number and alternating groups

Flag-transitive 2-designs with prime square replication number and alternating groups

Pentavalent Cayley graphs of order twice a prime square with solvable automorphism groups

A Cayley graph [Formula: see text] on a finite group [Formula: see text] is said to be normal if the right regular representation [Formula: see text] of [Formula: see text] is normal in the full automorphism group of [Formula: see text]. In this paper, we classify all connected pentavalent non-normal Cayley graphs of order [Formula: see text] with solvable automorphism groups and [Formula: see text] a prime.

The Basic Locally Primitive Graphs of Order Twice a Prime Square

A graph Γ is called G-basic if G is quasiprimitive or bi-quasiprimitive on the vertex set of Γ, where G⩽Aut(Γ). It is known that locally primitive vertex-transitive graphs are normal covers of basic ones. In this paper, a complete classification of the basic locally primitive vertex-transitive graph of order 2p2 is given, where p is an odd prime.

On prime-valent symmetric graphs of order four times an odd prime power

In this paper, we classify symmetric basic prime-valent graphs of order four times an odd prime power. As an application, prime-valent symmetric graphs of order four times a prime or prime square are specifically determined. We remark that special cases for valency 5 and 7 were characterized by Yang etc. (Eur. J. Combin. 80:236–246, 2019) and Pan and Yin (J. Algebr. Appl. 17(5):1850093, 2018).

The wild McKay correspondence for cyclic groups of prime power order

The v-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical. With this criterion, we obtain more examples of quotient varieties which are Kawamata log terminal (klt) but not Cohen–Macaulay.

Paramedial quasigroups of prime and prime square order

We prove that, for every odd prime number [Formula: see text], there are [Formula: see text] paramedial quasigroups of order [Formula: see text] and [Formula: see text] paramedial quasigroups of order [Formula: see text], up to isomorphism. We present a complete list of those which are simple.

Squared prime numbers

Squared prime numbers

Flag-transitive 2-designs with the prime square replication number

In this note, we show that if G is a flag-transitive automorphism group of a symmetric 2-design D with the prime square replication number, then G is point-primitive of affine or almost simple type. If D is a non-symmetric 2-design and G is a primitive group, then G is of affine, almost simple or product type with G≤H≀S2 acting on P=Δ×Δ, where H is a primitive group on Δ.

Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’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 N−1, N/2 . As N 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.

Hopf Galois structures on field extensions of degree twice an odd prime square and their associated skew left braces

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

A pair of equations in unlike powers of primes and powers of 2

Abstract In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.

A classification of tetravalent non-normal Cayley graphs of order twice a prime square

A Cayley graph \(\mathrm{Cay}(G, S)\) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of \(\mathrm{Cay}(G, S)\). In this paper, a complete classification is given of tetravalent non-normal Cayley graphs of order \(2p^2\) for each prime p.

An L-function free proof of Hua's Theorem on sums of five prime squares

Abstract We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.