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 formula for the number of primes p = q⌊αn + β⌋ + a with n ≤ N, where α,β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a,q are integers with $0 ≤ a < q ≤ N^κ$ and gcd(a,q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a(mod q).
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We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if $ℬ_{2ℓ}$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on $ℬ_{2ℓ}$.
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Let φ(·) and σ(·) denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of m ≤ x for which the equation m = σ(n) - n has no solution. We also show that the set of positive integers m not of the form (p-1)/2 - φ(p-1) for some prime number p has a positive lower asymptotic density.
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Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence $ℬ_{α,β} = (⌊αn + β⌋)_{n=1}^{∞}$. We conjecture that the same result holds true when α is an irrational number of infinite type.
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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $𝓗(x) = {gx + h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $𝓗(n) = {gn + h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $𝓗(n) = {gn + h_j}_{j=1}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_1,...,δ_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $δ_{j-1} | δ_j$ for 2 ≤ j ≤ m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.
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