Numerical calculation of the density of prime numbers with a given least primitive root

In this paper the densities of prime numbers having the least primitive root , where is equal to one of the initial positive integers less than 32, have been numerically calculated. The computations were carried out under the assumption of the Generalised Riemann Hypothesis. The results of these computations were compared with the results of numerical frequency estimations.

     

Comments on search procedures for primitive roots

Let be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct residues modulo , one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm.

     

On the cycle structure of repeated exponentiation modulo a prime

In a recent work, Shallit and Vasiga have obtained several results about tails and cycles in orbits of repeated squaring. Some of these results have been based on the Extended Riemann Hypothesis. Here, we extend their result to repeated exponentiation with any fixed exponent e and also show that in fact classical unconditional results about the distribution of primes in arithmetic progressions, combined with very elementary arguments, are quite sufficient to generalise and give an unconditional proof of their asymptotic formulas.     

On prime representing polynomials

A heuristic method is presented to determine the number of primesp ≤x, represented by an irreducible polynomialf(n), without non-trivial fixed factor (f(y)<∈Z[y]; n∈Z. The method is applied to two specific polynomials and the results are compared with those of the heuristic approach of Hardy and Littlewood.     

On the least primitive root in number fields

Let K be an algebraic number field and $ \mathfrak{O} $ _K its ring of integers. For any prime ideal $ \mathfrak{p} $ , the group $ (\mathfrak{O}_K /\mathfrak{p})* $ of the reduced residue classes of integers is cyclic. We call any element of a generator of the group $ (\mathfrak{O}_K /\mathfrak{p})* $ a primitive root modulo $ \mathfrak{p} $ . Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal $ \mathfrak{p} $ under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.     

On short Kloosterman sums modulo a prime

Using the Karatsuba method, we obtain new estimates for Kloosterman sums modulo a prime, which, under certain constraints on the number of summands, are sharper than similar estimates found earlier.

     

Gaps between integers with the same prime factors

We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers is , and it is shown that this lower bound is best possible. Dressler's conjecture is verified for values of and up to .

     

On the distribution of the primitive roots of a prime

In this paper it is shown that the number of pairs of consecutive primitive roots modulo p is asymptotic to $\text{(p ? 2)(}\text{?(p ? 1)}\text{(p ? 1)}\text{)}{}^2$, and that, for all sufficiently large primes p, there is at least one pair of consecutive primitive roots modulo p. The theorem proved here is a generalization of this proposition. Another one is mentioned in the remarks.     

A spectral interpretation for the explicit formula in the theory of prime numbers

A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.     

Classification theorems for sumsets modulo a prime

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions:(1) When can one represent zero as a sum of some elements of A?(2) When can one represent every element of Zp as a sum of some elements of A?(3) When can one represent every element of Zp as a sum of l elements of A?     

Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and denote the number of solutions of the congruence . In this paper, using the third-order recurring sequences we determine the values of N_p(x³+a₁x²+a₂x+a₃) and N_p(x⁴+ax²+bx+c), and construct the solutions of the corresponding congruences, where a₁,a₂,a₃,a,b,c are integers.     

On non-congruent numbers with 1 modulo 4 prime factors

In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include Li and Tian’s result as special cases.     

On the -rank Artin Conjecture

We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which can be generated by given multiplicatively independent numbers. In the case when the 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 .

     

On the least prime in an arithmetic progression

Letq, k be positive integers, (q, k)=1, andP(k, q) be the smallest prime number satisfyingpэk (modq) In this paper, we prove that forq sufficiently large, one hasP (k,q)≪q ⁸.     

Waring type congruences involving factorials modulo a prime

In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form