Showing papers on "Prime number published in 1997"
TL;DR: It is reported that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson prime p < 5x 10 8 .
Abstract: An odd prime p is called a Wieferich prime if 2 P-1 = 1 (mod p 2 ) alternatively, a Wilson prime if (p - 1)|= -1 (mod p 2 ). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5,13, and 563. We report that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson primes p < 5x 10 8 . It is elementary that both defining congruences above hold merely (mod p), and it is sometimes estimated on heuristic grounds that the probability that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p).
146 citations
TL;DR: In this article, it was shown that if the generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers, and the proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and extensive computation.
Abstract: We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation
58 citations
TL;DR: The generalized Riemann hypothesis is assumed and an asymptotic formula for the number of primes for which F p * can be generated by r given multiplicatively independent numbers is proved.
Abstract: We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which F p * can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to 9 10 4 .
36 citations
TL;DR: In this article, two formulae for the number of sublattices of a given index k of an n-dimensional lattice are presented based on the decomposition of the index k into a product of prime numbers and have the form of a rational function of these primes.
Abstract: Two formulae for the number of sublattices of a given index k of an n-dimensional lattice are presented. They are based on the decomposition of the index k into a product of prime numbers and have the form of a rational function of these primes. Compared with other known methods, they can give the result in a much quicker and more comfortable way.
33 citations
TL;DR: A precise conjecture about the value of the Ginfinity-Euler characteristic of the Selmer group of E over Finfinity is made about the behavior of thisSelmer group as a module over the Iwasawa algebra.
Abstract: Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.
30 citations
26 citations
TL;DR: In this article, it was shown by combining the argument of Brudern with the reversal role technique in the sieve theory that one can replace the subscript 4 by 3, where r is any fixed positive number.
Abstract: Let P s denote the natural numbers that are the product of at most s prime numbers, and let p, q, r denote prime numbers. In connection with the Waring-Goldbach problem for cubes, J. Brudern proved that almost all numbers n are written in the form n = P 4 3 + p 3 + q 3 + r 3 (Ann. Scient. Ec. Norm. Sup., 1995). In this note, it is shown by combining the argument of Brudern with the reversal role technique in the sieve theory that one can replace the subscript 4 by 3. More precisely, all n≤N with some local conditions, except for O (N ( log N ) -A ) exceptions, can be written in the form n = P 3 3 + p 3 + q 3 + r 3, where A is any fixed positive number. This yields at once that every sufficiently large even number can be written as a sum of cubes of seven primes and a P 3.
21 citations
Book•
7 Mar 1997
TL;DR: In this article, the Division Algorithm Rational Integers Normal Integers Norms Gaussian Numbers Q (v2) Polynomials Notes Chapter 3. The Euclidean Algorithm Bezout's Equation Relatively Prime Numbers Gaussian Integers Notes Chapter 4. Units Elementary Properties Bezouts Equations Wilson's Theorem Order of Elements: Fermat and Euler Quadratic Residues Z [v2] Notes Chapter 5. Unique Factorization into Primes Zn Notes Chapter 6.
Abstract: Chapter 1. New Numbers A Planeful of Integers, Z[i] Circular Numbers, Zn More Integers on the Number Line, Z [v2] Notes Chapter 2. The Division Algorithm Rational Integers Norms Gaussian Numbers Q (v2) Polynomials Notes Chapter 3. The Euclidean Algorithm Bezout's Equation Relatively Prime Numbers Gaussian Integers Notes Chapter 4. Units Elementary Properties Bezout's Equations Wilson's Theorem Orders of Elements: Fermat and Euler Quadratic Residues Z [v2] Notes Chapter 5. Primes Prime Numbers Gaussian Primes Z [v2] Unique Factorization into Primes Zn Notes Chapter 6. Symmetries Symmetries of Figures in the Plane Groups The Cycle Structure of a Permutation Cyclic Groups The Alternating Groups Notes Chapter 7. Matrices Symmetries and Coordinates Two-by-Two Matrices The Ring of Matrices M2(R) Units Complex Numbers and Quaternions Notes Chapter 8. Groups Abstract Groups Subgroups and Cosets Isomorphism The Group of Units of a Finite Field Products of Groups The Euclidean Groups E (1), E (2), and E (3) Notes Chapter 9. Wallpaper Patterns One-Dimensional Patterns Plane Lattices Frieze Patterns Space Groups The 17 Plane Groups Notes Chapter 10. Fields Polynomials Over a Field Kronecker's Construction of Simple Field Extensions Finite Fields Notes Chapter 11. Linear Algebra Vector Spaces Matrices Row Space and Echelon Form Inverses and Elementary Matrices Determinants Notes Chapter 12. Error-Correcting Codes Coding for Redundancy Linear Codes Parity-Check Matrices Cyclic Codes BCH Codes CDs Notes Chapter 13. Appendix: Induction Formulating the n-th Statement The Domino Theory: Iteration Formulating the Induction Statement Squares Templates Recursion Notes Chapter 14. Appendix: The Usual Rules Rings Notes Index
18 citations
17 citations
Posted Content•
TL;DR: In this paper, the authors give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s = n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence.
Abstract: Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which:
- The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l different from p, the geometric cycle map from algebraic cycles over X to the Galois invariants of the l-adic cohomology of the geometric fibre of X, tensored by Q_l, is surjective.
- For X as above, the algebraicity of the Kunneth components of the diagonal and the hard Lefschetz theorem for cycles modulo numerical equivalence.
- For X as above, the existence of a filtration conjectured by Beilinson on the Chow groups of X.
- The rational Bass conjecture: for any smooth variety X over F_p, the algebraic K-groups of X have finite rank.
- The Bass-Tate conjecture: for F a field of characteristic p, of absolute transcendence degree d, the i-th Milnor K-group of F is torsion for i>d.
- Soule's conjecture: given a quasi-projective variety over F_p, the order of the zero of its Hasse-Weil zeta function at an integer n is given by the alternating sum of the ranks of the weight n part of its algebraic K'-groups.
16 citations
TL;DR: Theoretical motivation for searching for Ramsey graphs of prime order is given and additional computational evidence that primes tend to be better than composites is provided.
Abstract: We present new explicit lower bounds for some Ramsey numbers. All the graphs are cyclic and are on a prime number of vertices. We give theoretical motivation for searching for Ramsey graphs of prime order and provide additional computational evidence that primes tend to be better than composites.
Patent•
7 Aug 1997
TL;DR: In this paper, the problem of generating a pseudo-random number whose internal state is difficult to be estimated by a third party at high speed with a circuit of a small scale is addressed.
Abstract: PROBLEM TO BE SOLVED: To achieve a device to generate a pseudo-random number whose internal state is difficult to be estimated by a third party at high speed with a circuit of a small scale. SOLUTION: T pieces of prime numbers P(0), P(1)... P(T) are preliminarily written in memory 103, when an output of a T-ary counter 101 is j, a prime number P(j) is outputted and the output is supplied to an extended affine transformation circuit 104. Extended affine transformation EA,B,P(j) is performed to n-bits held in a register 102 depending on the prime number p(j) to be supplied from the memory 103 and a transformation result is outputted to a reducing function circuit 105 and the register 102 is simultaneously updated according to the transformation result by the extended affine transformation circuit 104. A reducing function is performed to the output of the extended affine transformation circuit 104 and s-bits (s
TL;DR: In this paper, two theorems on the p-divisibility of the parameters of a cyclic difference list, where p is a prime number, were proved and extended to the cyclic group of order 8.
TL;DR: In this article, it was shown that the set of elements of Cl (M ) which are realizable by some tamely ramified extensions with Galois groups isomorphic to Θ using some Stickelberger elements, and proved that it is a subgroup of Cl(M ).
TL;DR: Borders on the number of pairs (P, Q) with 0 ≤ P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters are given.
Abstract: We give bounds on the number of pairs (P, Q) with 0 ≤ P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q).
TL;DR: Balog and Harman as mentioned in this paper used the method of Balog, Harman and Kaufman to prove a theorem of Bombieri-Vinogradov type for prime numbers from the set Sλ.
Abstract: A different approach to this problem was developed by Linnik [11] in 1945. In 1979 Kaufman [10] used the method of Linnik and proved the asymptotic formula (1) for λ < 0.1631 . . . He also proved that if the Riemann Hypothesis is assumed then (1) holds for λ < 1/4. In 1983 Balog [1] and Harman [8] used Vaughan’s identity and mean value estimates for Dirichlet polynomials and independently proved without assuming the Riemann Hypothesis that the formula (1) is true for λ < 1/4. Later Balog [2] generalized his result to prime numbers in arithmetic progressions. We should also mention the works of Schoissengeier [14], [15], Gritsenko [7] and Rivat [13]. In the present paper we use the method of Balog and Harman and we prove a theorem of Bombieri–Vinogradov type for prime numbers from the set Sλ. Let λ, θ be real numbers such that
TL;DR: An upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors is obtained.
Abstract: We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors.
TL;DR: It is proved that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Zqn (or Zn) for any given integer q ⩾ 2 is polynomial time learnable using at most n + 1 equivalence queries.
TL;DR: In this article, an algorithm for computing the Selmer groups corresponding to 2-isogenies of elliptic curves over Q defined by the equation Ez : y2 = (x+ z)(x2 + z2) with arbitrary z ∈ Q∗ is presented.
Abstract: In their article [S-T], Roel J. Stroeker and Jaap Top considered elliptic curves over Q defined by the equation Ep : y2 = (x+ p)(x2 + p2), where p ∈ P is a prime number. They determined the Selmer groups corresponding to certain 2-isogenies and the sign of the functional equation of these curves. Moreover, they gave a method for computing the Mordell–Weil group Ep(Q) in some cases. The aim of this note is to generalize their method to curves over Q defined by the equation Ez : y2 = (x+ z)(x2 + z2) with arbitrary z ∈ Q∗. Whereas the methods are analogous to those of R. J. Stroeker and J. Top, the results obtained here are quite different. We shall develop an algorithm for computing the Selmer groups corresponding to 2-isogenies of these curves. This algorithm is based on four theorems, which constitute the main results of Section 2 and describe the Selmer groups of these curves. We also generalize the procedure for finding generators of the Mordell–Weil groups of these curves. This procedure terminates if the Tate–Shafarevich groups are trivial, which is certainly not so in general. Stroeker and Top were able to prove that the Tate–Shafarevich group is nontrivial in a special case of a prime k = p ≡ 9 mod 16 and ( 1+ √−1 p ) = 1 (see [S-T]), but I could not generalize their method.
Patent•
15 Dec 1997
TL;DR: In this paper, the problem of reducing a necessary memory capacity by sequentially generating larger prime numbers based on a prime number not exceeding 2n bits and generating a second prime number having a specified bit length was solved.
Abstract: PROBLEM TO BE SOLVED: To reduce a necessary memory capacity by sequentially generating larger prime numbers based on a prime number not exceeding 2n bits and generating a second prime number having a specified bit length. SOLUTION: A random number generation part 5 generates an odd number having double bit length of a word size, and an arithmetic operation part 4 performs division for trial by using a prime number not larger than word size of the prime number table. When a generated odd number cannot be divided by all the prime numbers in the prime number table, this odd number is judged as a prime number. A parameter (i) of the following processing loop is set to zero (0). Based on a prime number Fi , a random number generation part 5 generates a random number Ri satisfying Ri
TL;DR: In this article, the authors considered the continuous semisimple representation ρf : Gal(Q/Q) → GL(2,F`) associated to f by the work of Shimura, Deligne, and Deligne-Serre.
Abstract: The conjecture of Serre referred to in the title is the one relating Galois representations to modular forms [12]. Let Q denote the field of algebraic numbers in C and Z the subring of algebraic integers. Fix a prime number ` and a prime ideal l of Z lying over `, and put F` = Z/l. Given a cuspidal Hecke eigenform f(z) = ∑ n>1 a(n)e 2πinz of level N , weight k, and character , we consider the continuous semisimple representation ρf : Gal(Q/Q) → GL(2,F`) associated to f by the work of Shimura (k = 2), Deligne (k > 2), and Deligne-Serre (k = 1). It is characterized up to isomorphism by the formulas
TL;DR: In this paper, it was shown that the Picard group of a commutative noetherian ring can be constructed from functors of the form B \mapsto M tensor_A B, where M is a f.
Abstract: Let A be a commutative noetherian ring. Call a functor > --> > coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When such a functor F in fact takes its values in >, we show that there are only finitely many prime numbers p such that _p F(A) is infinite, and that none of these primes are invertible in A. This (and related statements) yield information about torsion in Pic(A). For example, if A is of finite type over Z, we prove that the torsion in Pic(A) is supported at a finite set of primes, and if _p Pic(A) is infinite, then the prime p is not invertible in A. These results use the (already known) fact that if such an A is normal, then Pic(A) is finitely generated. We obtain a parallel result for a reduced scheme X of finite type over Z. We show that the groups which can occur as the Picard group of a scheme of finite type over a finite field all have the form (finitely generated) + sum_{n=1}^infty F, where F is a finite p-group. Hard copy is available from the author. E-mail to jaffe@cpthree.this http URL.
TL;DR: In this paper, the congruence relation modulo p extended to the ring of rational numbers with denominators not divisible by p was considered and the residue class of m/n-r/s (mod p) was shown to be the residue of m times the inverse of n in Zp.
Abstract: Congruences for prime numbers p have always been of great interest. Examples include Fermat's Little Theorem (nP _ n (mod p)) or Wilson's theorem ((p-1)!--1 (modp)). In the following we consider the congruence relation modulo p extended to the ring of rational numbers with denominators not divisible by p. For such fractions m/n-r/s (mod p) if and only if ms-nr (mod p), and the residue class of m/n is the residue class of m times the inverse of the residue class of n in Zp. The purpose of this note is to state and prove the following result. Theorem. Let p be an odd przme. Then
Dissertation•
1 Jan 1997
TL;DR: In this paper, Guiga and Lehmer's conjecture was shown to hold for up to 14 prime factors of any counterexample of the form n-1 + 1.
Abstract: We call a family of primes $P$ normal if it contains no two primes $p,q$ such that $p$ divides $q-1$. In this thesis we study two conjectures and their related variants. Guiga's conjecture is that \sum_{k=1}^{n-1} k^{n-1} \equiv n-1 \mod {n)$ implies $n$ is prime. We study a group of eight variants of this equation and derive necessary and sufficient conditions for which they hold. Lehmer's conjecture is that $\phi(n)|n-1$ if and only if $n$ is prime. This conjecture has been verified for up to 13 prime factors of $n$, and we extend this to 14 prime factors. We also examine the related condition $\phi(n)|n+1$ which is known to have solutions with up to 6 prime factors and extend the search to 7 prime factors. For both of these conjectures the set of prime factors of any counterexample $n$ is a normal family, and we exploit this property in our computations.
1 Jan 1997
TL;DR: In this paper, it was proved that ClZ[zeta(n)] is a direct summand of Pic(Z[C-pn]), where p is an odd prime number, zeta n = 1 and C-pn is the cyclic group of order p(n).
Abstract: It is proved that ClZ[zeta(n)] is a direct summand of Pic(Z[C-pn]), where p is an odd prime number, zeta(n)(pn) = 1 and C-pn is the cyclic group of order p(n).
1 Aug 1997
TL;DR: In this article, a lower bound for the size of the largest prime divisor of an expression of the type axY + byx, where ab f = 0, x, y 2: 2 and gcd (x, y) = l.
Abstract: In [3] and [5] the authors ask how many primes are of the form xY + yX, where gcd (x, y) = 1 and x, y 2: 2. Moreover, Jose Castillo (see [2]) asks how many primes are of the Smarandache form xil + X2 X3 + ... + Xn Xl , where n > 1, Xl, X2, ••• , Xn > 1 and gcd (Xl, X2, ••• , X n ) = 1 (see [9]). In this article we announce a lower bound for the size of the largest prime divisor of an expression of the type axY + byx, where ab f= 0, x, y 2: 2 and gcd (x, y) = l. For any finite extension F of Q let dF = [F : Q]. For any algebraic number ( E F let NF (() denote the norm of (. For any rational integer n let Pen) be the largest prime number P diViding n with the convention that P(O) = P(±l) = l.
TL;DR: In this paper, the discrimination between primes of the type 1(mod 4) and of the kind 1 (mod 4 + 4) has been studied, and it has been shown that there are sulrrisingly many differences in behavior between the two families.
Abstract: We often come across examples of how nature discriminates among its children-bestowing wonderful favors on some while others seem less fortunate. The following note documents one such case study, namely, the discrimination between primes of the type 1(mod 4) and of the type 1(mod 4). We shall find that there are sulrrisingly many differences in behavior between the primes of these two families. Our main aim will be to present a proof of the following result (see [3], page 113), which does not seem to be too well-known: if p is a prime number of the form 1(mod 4), then
TL;DR: This work succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression, made possible because of the increase in computer capability and availabiblity, and the ability to obtain computational help via the Internet.
Abstract: In 1967 the first set of 6 consecutive primes in arithmetic progression was found In 1995 the first set of 7 consecutive primes in arithmetic progression was found Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression This was made possible because of the increase in computer capability and availabiblity, and the ability to obtain computational help via the Internet Although it is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression, it is very likely that 10 primes will remain the record for a long time