Narrow Results

A short review of Schrödinger Hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose–Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach.

Several mathematical frameworks of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find that they are related to the Riemann zeta function and to incomplete Gauss sums, respectively.

It is said that the numerical generation of exact chaotic time series by iterating, for example, the logistic map, will be impossible, because chaos has a high dependency on initial values. In this letter, an algorithm to generate them without the accumulation of inevitable round-off errors caused by the iteration is proposed, where rational numbers are introduced. Also, it is shown that the period of the chaotic time series depends on the rational numbers including large prime numbers, which are fundamentally related to the Mersenne and the Fermat prime ones. Since the time series are numerically regenerated by the proposed algorithm in an usual computer environment, it could be applied to cryptosystems which do not need the synchronization, and have a large key-space by using large prime numbers.

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.

In this paper, we give some heuristics suggesting that if (u_n)_n≥0 is the Lucas sequence given by u_n = (aⁿ - 1)/(a - 1), where a > 1 is an integer, then ω(u_n) ≥ (1 + o(1))log n log log n holds for almost all positive integers n.

Let c be a real number with 1 < c < 2. We study the representations of a large integer n in the form

A natural number n can generally be written as a sum of m consecutive natural numbers for various values of m ≥ 1. The length spectrum of n is the set of these admissible m. Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.

For n ≥ 1, let p_n be the nth prime number and for n ≥ 1. Using several results of Erdős, we study the sequence (q_n)_{n ≥ 1} and we prove similar results as for the sequence (d_n)_{n ≥ 1}, d_n = p_n+1 - p_n. We also consider the sequence for n ≥ 1 and denote by G_n and A_n its geometrical and arithmetical averages. We prove that for n ≥ 4.

We prove that a prime factor q of an odd perfect number x satisfies the inequality q < (3x)^1/3.

Let a be a natural number greater than 1. For each prime p, let i_a(p) denote the index of the group generated by a in . Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of i_a(p) is constant. We prove that the average value of i_a(p) is constant without using Conjecture A of Hooley. More precisely, we show upon GRH that for any α with 0 ≤ α < 1, there is a positive constant c_α > 0 such that

Let n ≥ 3 be an integer. For any permutation σ of A = {0, 1, …, n-1}, set . It is proved that n is prime if and only if 〈σ, n〉 ≡ 0 (mod n) for any σ. When n is composite, we show that there exist cycles of any length satisfying this congruence. We show as well that the analog statement is true for cycles of length at least 2 not satisfying the congruence.

We provide irreducibility conditions for polynomials of the form f(X) + p^kg(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g - deg f and p^k exceeds a certain bound depending on the coefficients of f and g, then f(X) + p^kg(X) is irreducible over ℚ.

We study the asymptotic behavior of the sequence with general term consisting of the ratio A_n by G_n, the arithmetic and geometric means of the prime numbers p₁, p₂, …, p_n, respectively, in which, p_n denotes the nth prime number.

We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval for all x ≥ 2; this improves a result of Ramaré and Saouter. We then show that the constant 4/π may be reduced to (1 + ϵ) provided that x is taken to be sufficiently large. From this, we get an immediate estimate for a well-known theorem of Cramér, in that we show the number of primes in the interval is greater than for c = 3 + ϵ and all sufficiently large values of x.

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on $\mathbb{N}$ with the division order is an example of $T_0$-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on $\mathbb{N}$, the continuity is equivalent to the Darboux property.

In RSA cryptography numbers of the form $pq$, with $p$ and $q$ two distinct proportional primes play an important role. For a fixed real number $r>1,$ we formalize this by saying that an integer $pq$ is an RSA-integer if $p$ and $q$ are primes satisfying $p<q\le rp$. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3(\mathrm{\text{mod}}4)$. In this paper, we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers $\le x$ which is much more precise than an earlier one derived by Decker and Moree in 2008.

Spiro proved that the identity function is the only multiplicative function with $f(p)\ne 0$ for some prime $p$ and $f(p+q)=f(p)+f(q)$ for all prime $p$ and $q$. We determine the sets $S$ of primes for which restricting our condition to $f(p+q)=f(p)+f(q)$ for all $p,q\in S$ still implies that $f$ is the identity function. We prove that $S$ satisfies these conditions if and only if $S$ contains every prime that is not the larger element of a twin prime pair and $S$ contains $5$ or $7$.

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n\le x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n$th (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number $\mu$ is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order $\mu {log}^{\frac{3}{2}}\mu$. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

A sequence $\mathcal{𝒜}={(a_i)}_{i\ge 0}$ of strictly positive integers is said to be quasi-primitive if there are no three distinct terms $a_i,a_j$ and $a_k\in \mathcal{𝒜}$ such that $(a_i,a_j)=a_k$. Erdős conjectured that the sum $f(\mathcal{𝒜},0)\le f(\mathcal{𝒬},0)$, where $\mathcal{𝒬}$ is the sequence of all powers of prime numbers and $f(\mathcal{𝒜},h)={\sum }_{a\in \mathcal{𝒜}}$ $\frac{1}{a(loga+h)}$. Recently, Laib et al. proved that the analogous conjecture of Erdős $f(\mathcal{𝒜},h)\le f(\mathcal{𝒬},h)$ is false for $h\ge 4.92$ over the quasi-primitive sequence of semiprimes. In this paper, by constructing a family of quasi-primitive sequences from sequences of k-almost primes and the powers of the prime numbers, we extend this falsity up to $1.46\cdots$.