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1
Content available remote

Roughly squarefree values of the Euler and Carmichael functions

100%
Banks W., Luca F.
Acta Arithmetica
|
2005
|
tom 120
|
nr 3
211-230
2
Content available remote

Prime numbers with Beatty sequences

100%
Banks W., Shparlinski I.
Colloquium Mathematicum
|
2009
|
tom 115
|
nr 2
147-157
EN
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).
3
Content available remote

Average Value of the Euler Function on Binary Palindromes

100%
Banks W., Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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tom 54
|
nr 2
95-101
EN
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 $ℬ_{2ℓ}$.
4
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Arithmetic properties of numbers with restricted digits

100%
Banks W., Shparlinski I.
Acta Arithmetica
|
2004
|
tom 112
|
nr 4
313-332
5
Content available remote

Nonaliquots and Robbins numbers

100%
Banks W., Luca F.
Colloquium Mathematicum
|
2005
|
tom 103
|
nr 1
27-32
EN
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.
6
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Carmichael numbers composed of primes from a Beatty sequence

100%
Banks W., Yeager A.
Colloquium Mathematicum
|
2011
|
tom 125
|
nr 1
129-137
EN
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.
7
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Consecutive primes in tuples

81%
Banks W., Freiberg T., Turnage-Butterbaugh C.
Acta Arithmetica
|
2015
|
tom 167
|
nr 3
261-266
EN
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.
8
Content available remote

Representations of integers as sums of primes from a Beatty sequence

81%
Banks W., Güloğlu A., Nevans C.
Acta Arithmetica
|
2007
|
tom 130
|
nr 3
255-275
9
Content available remote

Coincidences in the values of the Euler and Carmichael functions

71%
Banks W., Friedlander J., Luca F., Pappalardi F., Shparlinski I.
Acta Arithmetica
|
2006
|
tom 122
|
nr 3
207-234
10
Content available remote

Piatetski-Shapiro sequences

71%
Baker R., Banks W., Brüdern J., Shparlinski I., Weingartner A.
Acta Arithmetica
|
2013
|
tom 157
|
nr 1
37-68
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