Positive integers such that .
Positive integers such that .
Potenzfreie Werte von Polynomen in algebraischen Zahlkörpern.
Power Series with Multiplicative Coefficients.
Power-free values, large deviations, and integer points on irrational curves
Let be a polynomial of degree without roots of multiplicity or . Erdős conjectured that, if satisfies the necessary local conditions, then is free of th powers for infinitely many primes . This is proved here for all with sufficiently high entropy.The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.
Powerful values of quadratic polynomials.
Poznámka o číslech spřízněných
Pretentiousness in analytic number theory
In this report, prepared specially for the program of the XXVième Journées Arithmétiques, we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness” to help us better understand several key questions in analytic number theory.
Prime constellations in triangles with binomial coefficient congruences
The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by...
Prime divisors of polynomials at consecutive integers.
Prime divisors of some shifted products.
Prime facto rs of consecutive integers.
Prime factors of with an irreducible polynomial .
Prime factors of values of polynomials
We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.
Prime geodesic theorem.
Prime number theorem analogues without primes.
Prime numbers along Rudin–Shapiro sequences
For a large class of digital functions , we estimate the sums (and , where denotes the von Mangoldt function (and the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.
Prime numbers in short intervals
Prime numbers of the form p = m²+n²+1 in short intervals
Prime numbers with Beatty sequences
A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...