Elements of number theory as a mathematical basis for solving problems for processing integer numbers information

The article describes the methodology for preparing schoolchildren to solve problems for processing integer numbers information, where the use of a brute force algorithm is not rational. The statements of number theory necessary to solve the problems of fnding numbers with a given number of divisors of a certain form are formulated; their evidence is given. A set of exercises is presented that promotes the independent "discovery" of the listed statements by schoolchildren and the formation of a skill to identify the structure of a number with given properties. At the end of the article, algorithms and programs are presented to solve problem No. 25, presented in the training control and measuring materials of the Unifed State Exam in informatics in the 2020/2021 academic year. The exercise sets described in the article can be useful to both informatics teachers and schoolchildren preparing for the Unifed State Exam in informatics.

THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

On the bivariate Erdős–Kac theorem and correlations of the Möbius function

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and $u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If $y = x^{{1 \over \beta }}$, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.

A note on some congruences involving arithmetic functions

International audience We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.

NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.