Abstract: Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the square-free numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC/sup 0/ [p] for any prime p. Similar lower bounds are presented for the set of square-free numbers, and for the problem of computing the greatest common divisor of two numbers.

Abstract: We discuss the methods and results of a search for certain types of prime clusters. In particular, we report specific examples of prime 16-tuplets and Cunningham chains of length 14.

Abstract: In 1964, Ronald Graham proved that there exist relatively prime natural numbers $a$ and $b$ such that the sequence $\{A_n\}$ defined by $$ {A}_{n} =A_{n-1}+A_{n-2}\qquad (n\ge 2;A_0=a,A_1=b)$$ contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth's computation, Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham's construction and generalize Wilf's note, and show that the 12-digit pair $$(a,b)= (407389224418,76343678551)$$ also defines such a sequence.

Abstract: Suppose that N is a prime number greater than 19 and that P is a point on the modular curve X0(N) whose image in Jo(N) (under the standard embedding ι: X0(N)→J0(N)) has finite order. In [2], Coleman-Kaskel-Ribet conjecture that either P is a hyperelliptic branch point of X0(N) (so that N∈{23,29,31,41,47,59,71}) or else that ι(P) lies in the cuspidal subgroup C of J0(N). That article suggests a strategy for the proof: assuming that P is not a hyperelliptic branch point of X0(N), one should show for each prime number l that the l-primary part of ι(P) lies in C. In [2], the strategy is implemented under a variety of hypotheses but little is proved for the primes l=2 and l=3. Here I prove the desired statement for l=2 whenever N is prime to the discriminant of the ring End J0(N). This supplementary hypothesis, while annoying, seems to be a mild one; according to W.A. Stein of Berkeley, California, in the range N<5021, it is false only in case N=389.

Abstract: Using the techniques of automata theory, we give another proof of the function field analogue of the Mahler-Manin conjecture and prove transcendence results for the power series associated to higher divisor functions a(")= Edlndk. Let p be a prime number, and let lc be an algebraic closure of Fp. Let q be a variable and consider ad, a, E lc[[q]] defined by Theorem. The period q of the Tate elliptic curve y2 + xy = x3 + a42 + a, over K := lc(a4, a,) is transcendental over K. This function field analogue of the Mahler-Manin conjecture was proved by Voloch [V96] (soon afterwards the original conjecture was proved in [BDGP96]; see also [W96]), by approximating q by algebraic quantities and getting a contradiction by analyzing the Galois action using Igusa's theorem. We offer below a proof based on the automata criterion of algebraicity due to Christ01 [C79, CKMR801. Proof. It suffices to prove that a4 (resp. a,) is transcendental over lc(q), if p # 5 (resp. p = 5). Namely, the Hasse invariant of the Tate elliptic curve, i.e., the coefficient of xp-' in (x3 + x2/4 + a4x + a6)(p-')l2 for p > 3, is equal to one, which shows (essentially first noticed in [S-D731) that a4 and a, are algebraically dependent, for p > 3. (See the first part of the remarks below for the case p 2 and u is odd), then xnllnUqn/(lqn) is transcendental over Fp(q). Proof. With au(C) :=xdle du, we have Received by the editors August 27, 1997. 1991 Mathematics Subject Classification. Primary 11589, llG07, 68668, llB85.

Abstract: Nondegenerate σ-additive measures with ranges in ℝ and ℚq (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚq, where ℚq is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory.

Abstract: This note describes a logical method for the generalization of calculations, which is applied to Euler's factorization of the 5th Fermat prime, F 5 = 2 2 5 + 1.

Abstract: On the basis of a high degree of theory, prime numbers are derived through effective processing and steps so as to achieve a remarkable reduction in the processing time taken for the derivation. With respect to an arbitrary prime number rank entered, (1) numerical values are added in sequence to a prime number of the anterior rank to calculate prime number candidates of the next rank; (2) the thus calculated prime number candidate is divided by known prime numbers to verify whether it is a prime number or not; and (3) processing for reducing the verification time is iterated when the thus calculated prime number candidate is larger than a certain value, to thereby derive prime numbers until the entered rank is reached.

Abstract: Let p be an odd prime number. Let Λ = Zp((T )). We determine the Λ-isomorphism classes of finitely generated Λ-torsion Λ- modules with λ = 2and µ = 0 which have no non-trivial finite Λ- submodule. We apply this classification to Iwasawa modules X = lim − An associated to the cyclotomic Zp-extensions of imaginary qua- dratic fields and give some numerical examples.

Abstract: The problem of proving a number is of a given arithmetic format with some prime elements, is raised in RSA undeniable signature, group signature and many other cryptographic protocols. So far, there have been several studies in literature on this topic. However, except the scheme of Camenisch and Michels, other works are only limited to some special forms of arithmetic format with prime elements. In Camenisch and Michels’s scheme, the main building block is a protocol to prove a committed number to be prime based on algebraic primality testing algorithms. In this paper, we propose a new protocol to prove a committed number to be prime. Our protocol is O(t) times more efficient than Camenisch and Michels’s protocol, where t is the security parameter. This results in O(t) time improvement for the overall scheme.

Abstract: In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than 10000. For some starting values, it is shown for the first time that the sequence terminates. The current record for the maximum of a terminating sequence is located in the one starting at 4170; it converges to 1 after 869 iterations getting a maximum of 84 decimal digits at iteration 289.

Abstract: The complete generalized cycleG(d,n) is the digraph which hasZn×Zdas the vertex set and every vertex (i,x) is adjacent to thedvertices (i+ 1,y) withy?Zd. As a main result, we give a necessary and sufficient condition for the iterated line digraphG(d,n,k) =Lk?1G(d,n), withda prime number, to be a Cayley digraph in terms of the existence of a group?dof orderdand a subgroupNof (?d)nisomorphic to (?d)k. The condition is shown to be also sufficient for any integerd? 2. If?dis a ringRandNis a submodule ofRn, it is said thatG(d,n,k) is anR-Cayley digraph. By using some properties of the homogeneous linear recurrences in finite rings, necessary and sufficient conditions forG(d,n,k) to be anR-Cayley digraph are obtained. As a consequence, whenR=Zda new characterization for the digraphsG(d,n,k) to beZd-Cayley digraphs is derived. As a corollary, sufficient conditions for the corresponding underlying graphs to be Cayley can be deduced. Ifdis a prime power andFdis a finite field of orderd, the digraphsG(d,n,k) which areFd-Cayley digraphs are in 1-1 correspondence with the cyclic (n,k)-linear codes overFd.

Abstract: Let s be a fixed prime number, H a finite unramified extension of the field of rational s numbers Qe, and r a primitive q,~th root of unity, where qn = s if ~ r 2 and q,, = ~'*+2 if s = 2. We choose the roots (,, in such a way that the equality e (,,+1 = (,, holds for any n > 0. Then any finite cyclotomic extension of Qe is of the form k,, = H((,, ) for suitable H and n >_ -1 , where we put k-1 = H. Let U(k,~) be the group of units of k,,. Then the s logarithm defines a mapping log: U(k,.,) ) k,., (1) that is a group homomorphism. In [2] we studied the image log(U(kn)) C k,, and the properties of the pairing

Abstract: It is a beautiful property of prime numbers, first proved more than three centuries ago by Fermat, that k Pk (mod p) for all prime numbers p and all integers k. Here we present a simple proof of Fermat's "little" theorem by considering iterates of the function f(z) = z k on the complex plane. The method of proof has the advantage of generalizing the theorem to composite exponents: for every n we find a degree-n polynomial, with coefficients + 1, that is always divisible by n. This is different from Euler's generalization (k 4(n) =1 (mod n) for k and n copiime). The method of proof is potentially more general still, since it is easily adapted to other functions f. Indeed, for any set S, every function f: S -> S satisfying a certain property corresponds to a divisibility result similar to Fermat's little theorem. Let k be a positive integer and p be prime. Consider the function f(z) = zk for complex z. The pth iterate of f is evidently fP(z) = zk. Let Pp be the set of those z that are fixed under fP but not under f itself. Then IPp I = k P-k. But if z E Pp, then fJ(z) Epp for every i = 0,1,. . ., p 1; and since p is prime, the p values z,f(z),...jfP-(z) are all distinct. Hence, we can partition Pp into equivalence classes, each containing p elements, obtaining

Abstract: Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x1c1,⋯,ck) denote the number of primesp withp⩽x,\(p = [n_1^{c_1 } ] = \cdots [n_k^{c_k } ]\), where 1 k−k/(4k2+2).

Abstract: PROBLEM TO BE SOLVED: To provide a method and device for generating prime numbers and a storage medium with stored program for generating the prime numbers, permitting to assure uniqueness of mass-generated prime numbers and also reduce a memory quantity to be used therefor. SOLUTION: When it is assumed that k is a positive integer; m is a positive integer less than k; w is a positive integer less than 2m; and storage areas having lengths of k-bits and m-bits are prepared, a pre-given initial value is set in the m-bit storage area, and a progression whose period is w and whose value range is 1, 2, ..., 2m-1} is calculated, and when prime number generation is externally requested for via a telecommunication means, a value following the input value in the progression inputting the m-bit storage area as an input value; the m-bit value is copied in a prescribed place of the k-bit storage area; a random number of (k-m) bits is generated and copied in the remaining (k-m) bits of the k-bits storage area; the number is regarded as an integer and an prime number judgment is performed; and when it is a prime number, the k-bit data are returned outside via the telecommunication means, and when it is not a prime number, a random number is generated again.

Abstract: (1.1) N = p1 + p 2 2 + p 2 3 + p 2 4 + p 2 5 is solvable for large odd N satisfying N ≡ 5 (mod 24). This theorem can be regarded as a nonlinear extension of the Goldbach ternary theorem (Goldbach–Vinogradov Theorem), it also gives a deep insight into the Lagrange four square theorem. In this paper we study the equation (1.1) with prime variables in an arithmetic progression, i.e. the prime variables satisfy pi ≡ bi (mod d), i = 1, . . . , 5, and b = (b1, . . . , b5) ∈ B(N, d), where (1.2) B(N, d) = {b ∈ N5 : 1 ≤ bi ≤ d, (bi, d) = 1, b1 + . . .+ b5 ≡ N (mod σ(d)d)}, with σ(d) = 1, 4, 2 for 2 d, 2 ‖ d and 4 | d respectively. We will use this notation in the rest of the paper. Our main result is