Abstract: In this paper we shall investigate Giuga’s conjecture which asserts an interesting characterization of prime numbers, just as Wilson’s Theorem. Some variations and consequences of the Giuga congruence are discussed by means of Bernoulli numbers. In addition, we shall study various quotients relating to the integers satisfying the Giuga congruence.

Abstract: © Université Bordeaux 1, 1995, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: We give a small, but very useful modification of a criterion of Mignotte ([4]) for Catalan’s equation, replacing the class number of a certain abelian field by the relative class number, which is much easier to compute. The proof is the same, apart from the idea to consider the class group modulo the ideals coming from the real subfield. We use the following notation: K is a CM-field, IK its group of fractional ideals and i : K∗ → IK the canonical map x 7→ (x); j denotes complex conjugation, K+ the maximal real subfield and h−(K) the relative class number of K; OK is the ring of integral elements of K. Lemma 1. Let K be a CM-field and Q a finite set of prime ideals of K. There is a subgroup I0 of the ideal group IK such that (i) the prime ideals in Q do not appear in the factorization of any ideal in I0; (ii) IK/(i(K∗)I0) has cardinality h−(K) or 2h−(K); (iii) if e ∈ K∗ with (e) ∈ I0, then e1−j is a root of unity. P r o o f. Let I0 consist of those ideals which are in the image of the canonical map IK+ → IK , and which do not contain any prime ideal in Q. If (e) ∈ I0, then (e) = (e), so e1−j is a unit, hence also a root of unity because all its conjugates have absolute value 1 (cf. [6], Lemma 1.6). It remains to show (ii). It is an easy consequence of the approximation theorem that every ideal class contains an ideal without primes in Q (see e.g. [3], IV, Corollary 1.4). Therefore IK/(i(K∗)I0) = ideal class group of K modulo image of the ideal class group of K+. By [6], Theorem 10.3, at most 2 ideal classes of K+ become principal in K, so the statement follows. Theorem 1. Let p 6= q be odd prime numbers. Let ζ be a primitive p-th root of unity and K an imaginary subfield of L := Q(ζ). Catalan’s equation x − y = 1 has no nontrivial integral solution if q -h−(K) and pq−1 6≡ 1 mod q2.

Abstract: Given a set π of prime numbers, the π-partial characters of a finite group G are defined as the restrictions of the ordinary characters of G to the π-elements of G. In the case where G is solvable, or even just π-separable, these π-partial characters are under good control and they can be used to prove new results about the ordinary characters of such groups. In these lectures, several of the key results that form the foundation of the theory are presented without proof. These are then used to develop the theory and derive a number of applications.

Abstract: k=i is close to log n; this shows that the harmonic series diverges very, very slowly. For example, it takes more than 1.5 x 1043 terms for its partial sums to reach 100; see [3], [4], and [5]. In this paper, we refine and thin out the harmonic series to show that the divergence of this series depends on some of its specific terms and without those terms the remaining subseries is convergent. Then, an upper bound and an approximate value will be given to the value of the convergent subseries. Finally, by use of the derived results, it will be shown that the divergence of the Euler series E: (over prime numbers p) depends on specific prime numbers.

Abstract: Numbers of the forms C n = n.2 n + 1 and W n = n.2 n − 1 are both called Cullen numbers. New primes C n are presented for n = 4713, 5795, 6611, 18496. For W n , several new primes are listed, the largest one having n = 18885. Furthermore, all efforts made to factorize numbers C n and W n are described, and the result, the complete factorization for all n < 300, is given in a Supplement.

Abstract: The communication complexity of a function f denotes the number of bits that two processors have to exchange in order to compute f(x, y), when each processor knows one of the variables x and y, respectively. In this paper the deterministic communication complexity of sum-type functions, such as the Hamming distance and the Lee distance, is examined. Here f: X × X ? G, where X is a finite set and G is an Abelian group, and the sum-type function fn:Xn × Xn ? G is defined by fn((x1, ..., xn), (y1, ..., yn)) = ?ni=1f(xi, yi) Since the functions examined are also translation-invariant, their function matrices are simultaneously diagonalizable and the corresponding eigenvalues can be calculated. This allows to apply a rank lower bound for the communication complexity. The best results are obtained for G = Z/2Z. For prime numbers |X| in this case the communication complexity of all non-trivial sum-type functions is determined exactly. Exact results are also obtained for the parity of the Hamming distance and the parity of the Lee distance. For the Hamming distance and the Lee distance exact results are only obtained for special parameters n and |X|.

Abstract: A new criterion of the first case of Fermat Last Theorem is derived. This criterion is based on special sums of reciprocals.

Abstract: Prime numbers are the multiplicative building bricks of the number system. According to the fundamental theorem of arithmetic, every integer number larger than 1 is either a prime or the product of a unique set of primes. In what follows, by an integer we will understand a positive integer. In multiplicative number theory each integer is a word, more exactly a commutative juxtaposition of primes. In this coding process each prime is employed according to a rigid rule (the gap between the consecutive multiples of a prime p is just p) and the set of prime numbers is like an alphabet that is self-generating in order to make the resulting code nondegenerate. But how are the prime numbers themselves generated? Contemplating successive gaps between consecutive primes or the number of prime factors of consecutive integers, we can only notice an apparently chaotic behavior of the prime numbers leading us to believe that their distribution law must be very complicated. There are two different ways of looking at prime numbers: globally and algorithmically. From an algorithmic point of view the process of generating prime numbers is relatively clear. The prime-number sieve, attributed to the ancient Greek scholar Eratosthenes, was one of the first step-by-step methods invented for distinguishing primes from composites among the numbers up to some predetermined limit: Take the number 2, eliminate its multiples; the next prime is 3, eliminate its multiples; the next prime is 5, eliminate its multiples, etc. Today, checking whether or not an integer is a prime is one of the first computer programs learned in. any programming language. Eratosthenes' sieve simply tells us what to do, step-by-step, for selecting the primes in a given set of consecutive integers without revealing any regularity in the distribution of primes. Those unhappy with an algorithmic approach have tried several ways to approach a global understanding of the behavior of primes. Many papers have dealt with the asymptotic behavior of different functions depending on primes. There is a rich literature on the subject (see for instance [15], [17], [3], [2]) using very subtle mathematical techniques. To give only one example, let 7T(x) denote the number of primes not exceeding the positive real number x. According to the celebrated prime number theorem (PNT), we have 7T(x) = x/ln x, (x -> oo), which means that the ratio of the two functions, namely wr(x)/(x/ln x), converges to 1 as x grows without bound, proved independently by J. Hadamard [9] and C. J. de La Vallee Poussin [14] using tools involving functions of complex variables. PNT is a superb example of extracting asymptotic order from chaos.

Abstract: The goal of this paper is to discuss a Newton-Hodge inequality for modular forms. More precisely, for a prime number p and an integer N prime to p we consider the characteristic polynomial of the Hecke operator Up on the space Sk-^-2 (^i (^\" N)) of cusp forms for the congruence subgroup Fi (j»\" N) of SL'z (Z). The main theorem bounds the Newton polygon of this polynomial from below by an explicit polygon denned in terms of the genus and number of cusps of the modular curve Xi (N). The main technique is a motivic variation of theorems of Mazur, Ogus, Illusie and Nygaard on the Katz conjecture (according to which the Newton polygon of Frobenius on crystalline cohomology is bounded in terms of dimensions of Hodge cohomology groups) and a computation of these Hodge groups using logarithmic schemes. We get new information because the relevant Hodge nitration is not of type (k + 1,0), (0, k + 1) as usual, but rather of type (k + 1,0), (k, 1), ..., (1, k), (0, k + 1).

Abstract: Suppose that A = 〈A, …〉 is a first-order structure with a finite number of finitary relation and function symbols, and p is a prime number. We say that a function μ with mod p values is a mod p cardinality function if it is defined on the first-order definable subsets of A, A 2, … and it satisfies the following basic properties of the usual notion of cardinality. It is invariant under one-to-one first-order definable maps, it is additive with respect to disjoint union, it is multiplicative with respect to direct product, and the cardinality of singletons is 1. We show that if there is a first-order definable ordering of the universe then the existence of a mod p cardinality function is equivalent to the following statement: There are no two first-order definable equivalence relations Ф and Ψ on a (first-order definable) subset X of A i for some i = 1,2,… with the following properties: (1) each class of Ф contains exactly p elements, and (2) each class of Ψ with one exception contains exactly p elements, the exceptional class contains 1 element. (We will refer to this statement as the modulo p counting principle.)

Abstract: For an imaginary quadratic fieldK we study the asymptotic behaviour (with respect top) of the number of integers inK with norm of the formk(p−k) for some 1≤k≤p−1, wherep is a prime number. The motivation for studying this problem is that it is known by recent results due to G. Frey and E. Kani that knowledge of this asymptotic behaviour can lead to statements of existence of curves of genus 2 with elliptic differentials in particular cases.

Abstract: Primality testing of large numbers is very important in many areas of mathematics, computer science and cryptography, and in recent years, many of the modern primality testing algorithms have been incorporated in Computer Algebra Systems (CAS) such as Axiom and Maple as a standard. In this paper, we discuss primality testing of large numbers in Maple V Release 3, a Maple version newly released in 1994. Our computation experience shows that the Maple primality testing facility isprime, based on a combined use of a strong pseudoprimality test and a Lucas test, is efficient and reliable.

Abstract: PURPOSE:To provide a safe and sure signature, authentication and secret communication system using an elliptic curve. CONSTITUTION:An elliptic curve is as follows. (1) A finite body GF (p) being a definition body of an elliptic curve is set so that a class number of an imaginary second order body Q{(-d) } of a positive integer (d) is made small, a prime number (p) expressed as (2-alpha) (where, alpha is a small number) is set so that (p+1-a) or (p+1+a) can be divided by a large prime number and it is expressed as (p-a )=(d.b ) for the integer o. An elliptic curve and an element in which a root of a class polynomial Hd(x)=0 is a (j) constant are used on this finite body GF (p). (2) Elliptic curves E1, E2,... En of which the definition body is the definite body GF(p ) are constituted so that they are not the same type but the original numbers are equal, an used elliptic curve is properly changed.

Abstract: In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free groupwhere Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

Abstract: (2) |p1 + . . .+ p5 −N1| < e1(N1), |p1 + . . .+ p5 −N2| < e2(N2), where c and d are different numbers greater than one but close to one and e1(N1), e2(N2) tend to zero as N1 and N2 tend to infinity. Of course, we have to impose a condition on the orders of N1 and N2 because of the inequality (x1 + . . .+ x c 5) d/c ≤ x1 + . . .+ x5 ≤ 5(x1 + . . .+ x5) which holds for every positive x1, . . . , x5 provided 1 < d < c. We shall prove the following theorem.

Abstract: PROBLEM TO BE SOLVED: To provide a prime number generator, a prime factor discriminating device and a prime number generator having limitation in which prime numbers are generated having a high sefety so that any one of the bit numbers of the prime factors that should be included by P-1 and p+1 becomes less than 1/2 of the bit number of P and the appearance probability of the prime numbers is relatively uniform. SOLUTION: When a bit number pb of a prime number p, a prime number r which is the prime factor of p-1, a maximum value xx of x, a minimum value xm of x, a final value xs of x, a random number x of an even number in that xm<=x<=xx and a random number xd are externally inputted, compute p=xr+1 and input p into a prime number discriminator. If p is discriminated as a prime number, output p and the process is stopped. If p is discriminated as not a prime number, subtract 2 from x or add 2 to x in accordance with the value of xd. If x becomes less than xm, make x to be xx. If x becomes larger than xx, make x to be xm. If x becomes equal to xs, output a signal indicating that the prime number p which meets input conditions pb and r does not exist and stop. If x in not equal to xs, go back to the first process.

Abstract: In this paper, the prime number encoding scheme is proposed for representing the supervisory class signals at the output layer of multilayered feedforward networks. The notion of prime number encoding stems from the idea that the module representations with respect to two appropriately chosen primes can uniquely represent any bounded integer value. In particular, the derived prime number encoding scheme is a viable replacement for the conventional 1-per-class coding scheme used for indexing tasks. Face recognition experiments performed using a 24-face database, and word-spotting experiments performed with the TIMIT speech database, suggest that the prime number encoding scheme is beneficial for indexing purposes, not only from the point of parameter reduction, but also in terms of recognition performance.

Abstract: We discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$ are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is a positive integer. The technique used to prove Fermat's Last Theorem shows that the equation has no solutions with $\a>1$ or $b$ even. When $\a=1$ and $b$ is odd, there are the two trivial solutions $(\pm 1, \mp 1, \pm 1)$. In 1952, D\'enes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for $p\equiv1$ mod~4. We link the case $p\equiv3$ mod~4 to conjectures of Frey and Darmon about elliptic curves over~$\Q$ with isomorphic mod~$p$ Galois representations.

Abstract: Suppose that B is a sufficiently large positive constant, e is a sufficiently small positive constant, X and N are sufficiently large. It is mainly proved that i) for the positive integers n, Xn≤2X, except for O(X log-B X) values, the interval (n, n+n1/14+e) contains a prime number; ii) if A = N1/2+s, then the even numbers in the interval (N, N+A), except for 0(Alog-B N) values, are all Goldbach numbers.

Abstract: Since the discovery of the RSA encryption scheme, primality domain has gained much interest. For the generation of keys for this code, two prime numbers are used. Amongst the different methods to deal with this problem, we are here interested in generation of certified prime numbers and we present a new method less costly in terms of computation in regard of the other methods of generation for a given size of prime numbers.

Abstract: The parameters , of a generalized quadrangle are said to be classical if and for some prime number and nonnegative integers and . A description is obtained for the possible parameters of strongly regular graphs in which the neighborhoods of the vertices are generalized quadrangles with quasiclassical parameters.Bibliography: 6 titles.

Abstract: In the present paper, we will give a positive result relating to the l-part of Shioda’s problem [2] on Jacobi sums J (a) l (p) under a certain condition (see Corollary to Theorem 2 of the present paper), as an application of our congruence for Jacobi sums [1, Theorem 2] (see also Theorem 1 of the present paper). Let l be any prime number such that l ≥ 5, and let ζl be a primitive lth root of unity in C (the field of complex numbers). Let Q be the field of rational numbers and let Z be the ring of rational integers. Put k = Q(ζl). For any integer r ≥ 1 and any a = (a1, . . . , ar) ∈ Z and for any prime ideal p of k which is prime to l, let