Abstract: In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex circulant Hadamard matrices of size p, with diagonal entries equal to 1, is less or equal to (2p-2)!/(p-1)!^2.

Abstract: In this paper, we study the pro-Σ fundamental groups of configuration spaces, where Σ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via two somewhat distinct approaches, that, in many cases, the “fiber subgroups” of such fundamental groups arising from the various natural projections of a configuration space to lower-dimensional configuration spaces may be characterized group-theoretically.

Abstract: Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

Abstract: We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)−c(log log log log log n)) for some absolute c > 0.

Abstract: There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in ${\mathcal{C}}^{\infty}$ and is readily seen to be monotonic.

Abstract: Factorizing composite number n = qr, where q and r are two large primes, and finding discrete logarithm modulo large prime number p are two dicult computational problems which are usually put into the base of dierent digital signature schemes (DSSes). This paper introduces a new hard computational problem that consists in finding the kth roots modulo large prime p = Nk 2 + 1, where N is an even number and k is a prime with the length jkj ‚ 160. Diculty of the last problem is estimated as O( p k). It is proposed a new DSS with the public key y = x k mod p, where x is the private key. The signature corresponding to some message M represents a pair of the jpjbit numbers S and R calculated as follows: R = t k mod p and S = tx f(R;M) mod p, where f(R;M) is a compression function. The verification equation is S k mod p = y f(R;M) R mod p. The

Abstract: Combining the arguments developed in the works of D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim [Preprint, 2005, arXiv: math.NT/506067] and Y. Motohashi [Number theory in progress - A. Schinzel Festschrift (de Gruyter, 1999) 1053-1064] we introduce a smoothing device to the sieve procedure of Goldston, Pintz, and Yildirim (see [Proc. Japan Acad. 82A (2006) 61-65] for its simplified version). Our assertions embodied in Lemmas 3 and 4 of this article imply that a natural extension of a prime number theorem of E. Bombieri, J. B. Friedlander, and H. Iwaniec [Theorem 8 in Acta Math. 156 (1986) 203-251] should give rise infinitely often to bounded differences between primes, that is, a weaker form of the twin prime conjecture.

Abstract: Let p be an odd prime number, let E be an elliptic curve over a number field k, and let F/k be a Galois extension of degree twice a power of p. We study the Zp-corank rkp(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rkp(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, if G=Gal(F/K), if G+ is the subgroup of elements of G commuting with a choice of involution of F over k, and if rkp(E/K) is odd, then we show that (under mild hypotheses) rkp(E/F)≥[G:G+]. As a very specific example of this, suppose that A is an elliptic curve over Q with a rational torsion point of order p and without complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p such that every prime where both E and A have bad reduction has odd order in Fp× and such that the negative of the conductor of E is not a square modulo p, then there is a positive constant B depending on A but not on E or n such that rkp(E/Q(A[pn]))≥Bp2n for every n

Abstract: An integer t is a twin prime (see [5] or [6]), if t is a prime number ≥ 3 and if t − 2 or t + 2 is also a prime number ≥ 3 . Example. 1000000000061 and 1000000000063 are twin primes (see [6]). It is conjectured that there are infinitely many twin primes. The notion of a, friendly number (see [2] or [3] or [7] or [8] or [9] or [10]) is based on the idea that a human friend is a kind of alter ego. Indeed, Pythagoras wrote (see [8]): A friend is the other I , such as are 220 and 284 . These numbers have a special mathematical property: each is equal to the sum of the other’s proper divisors (divisors other than the number itself). The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 , and 110, and they sum to 284; the proper divisors of 284 are 1, 2, 4, 71 , and 142, and they sum to 220. So {220, 284} is called a pair of friendly numbers [[note {17296, 18416} is also a pair of friendly numbers (see [7] or [8])]]. More precisely, we say that a number a′ is a friendly number, if there exi...

Abstract: It is known that a $k$-term DNF can have at most $2^k - 1$ prime implicants and that this bound is sharp. We determine all $k$-term DNF having the maximal number of prime implicants. It is shown that a DNF is maximal if and only if it corresponds to a nonrepeating decision tree with literals assigned to the leaves in a certain way. We also mention some related results and open problems.

Abstract: In hyperelliptic curve cryptography, finding a suitable hyperelliptic curve is an important fundamental problem. One of necessary conditions is that the order of its Jacobian is a product of a large prime number and a small number. In the paper, we give a probabilistic polynomial time algorithm to test whether the Jacobian of the given hyperelliptic curve of the form Y 2 = X+uX+vX satisfies the condition and, if so, gives the largest prime factor. Our algorithm enables us to generate random curves of the form until the order of its Jacobian is almost prime in the above sense. A key idea is to obtain candidates of its zeta function over the base field from its zeta function over the extension field where the Jacobian splits.

Abstract: An n-Venn diagram is a Venn diagram on n sets, which is defined to be a collection of n simple closed curves (Jordan curves) C1,C2, ,Cn in the plane such that any two intersect in finitely many points and each of the 2n sets of the form ∩C i i is nonempty and connected, where i is one of “interior” or “exterior” Thus the Venn regions are all bounded except for the region exterior to all curves; each bounded region is the interior of a Jordan curve See [6] for much more information on Venn diagrams An n-Venn diagram is symmetric if each curve Ci is ρ i (C1), where ρ is a rotation of order n about some center (we use O for the fixed point of rotation ρ) We use Boolean notation for combinations of sets, with the 0-1 string e1e2 en representing ∩C i i , where i is interior (respectively, exterior) if ei = 1 (respectively, 0) Thus 111 1 represents F , the full intersection of all the interiors, 000 0 is the intersection of all the exteriors (the unbounded region), and 100 0 represents the set of points interior to C1 and exterior to the others In a symmetric Venn diagram, rotation of a region by ρ corresponds to a rightward cyclic shift of the Boolean string The universally familiar three-circle Venn diagram is symmetric, as is the one on two sets using two circles For about 40 years a major open question was whether symmetric n-Venn diagrams exist for all prime n Henderson found one for n = 5 and also (unpublished) for n = 7 Much later, Hamburger [3] settled the case of 11, which was quite complicated, and then in 2004 Griggs, Killian, and Savage [1] found an approach that works for all primes So we now have the strikingly beautiful theorem that a symmetric n-Venn diagram exists if and only if n is prime But there is a small problem: Henderson’s proof, which appears to be very simple, has a gap Here is the proof from [4] Suppose 1 ≤ k ≤ n − 1 Since a symmetric n-Venn diagram is symmetric with respect to a rotation of 2π/n, the regions corresponding to the Boolean strings with k 1s must come in groups of size n, each group consisting of one such region and its images under repeated rotation by 2π/n Therefore n divides (n k ) This concludes the proof because the only n for which this is true for the specified k-values are the primes (an easy-to-prove fact of number theory; see [5]) This is a very seductive argument The primeness arises in such a cute way that one wants it to be true Thus the proof has been repeated in many papers in the decades since it was first published Yet there are problems The proof does not call upon the connectedness of the Venn regions Without connectedness the result is false; see Figure 1 (due to Grunbaum [2]), which shows a diagram satisfying all of the conditions

Abstract: In this work, we show that the set of primes can be obtained through dynamical processes. Indeed, we see that behind their generation there is an apparent stochastic process; this is obtained with the combination of two processes: a “zig-zag” between two classes of primes and an intermittent process (that is a selection rule to exclude some prime candidates of the classes). Although we start with a stochastic process, the knowledge of its inner properties in terms of zig-zagging and intermittent processes gives us a deterministic and analytic way to generate the distribution of prime numbers. Thanks to genetic algorithms and evolution systems, as we will see, we answer some of most relevant questions of the last two centuries, that is “How can we know a priori if a number is prime or not? Or similarly, does the generation of number primes follow a specific rule and if yes what is its form? Moreover, has it a deterministic or stochastic form?” To reach these results we start to analyze prime numbers by using binary representation and building a hierarchy among derivative classes. We obtain for the first time an explicit relation for generating the full set P n of prime numbers smaller than n or equal to n .

Abstract: In this paper, we prove the following theorem: Let p be a prime number, P aS ylowp- subgroup of a group G and π = π(G) \{ p}. If P is seminormal in G, then the following statements hold :1 )G is a p-soluble group and P ≤ Op(G); 2) lp(G) ≤ 2 and lπ(G) ≤ 2; 3) if a π-Hall subgroup of G is q-supersoluble for some q ∈ π ,t henG is q-supersoluble. Keywords finite groups, seminormal subgroups, Sylow subgroups, p-soluble groups, p-supersoluble groups

Abstract: The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations, population dynamics, etc. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualises the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations, or an inter-connected network with associated long-range correlations. The model predictions are as follows: (i) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (ii) Fractals signify quantumlike chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (ii) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organisation of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.

Abstract: Dirichlet's proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Euler's earlier work on the zeta function and the distribution of primes. He first proves a simpler case before going to full generality. The paper was translated from German by R. Stephan and given a reference section.