298 research outputs found
On a conjecture of Pomerance
We say that k is a P-integer if the first phi(k) primes coprime to k form a
reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the
set of P-integers and conjectured that 30 is the largest P-integer. We prove
the conjecture assuming the Riemann Hypothesis. We further prove that there is
no P-integer between 30 and 10^11 and none above 10^3500.Comment: 10 pages. Submitted to Acta Arithmetic
Становлення і розвиток православних духовних семінарій на Правобережній Україні (кінець ХVІІІ – перша половина ХІХ ст.)
In this note we prove results of the following types. Let be given distinct complex numbers satisfying the conditions for and for every there exists an such that . Then If, moreover, none of the ratios with is a root of unity, then The constant −1 in the former result is the best possible. The above results are special cases of upper bounds for obtained in this paper
On the rationality of Cantor and Ahmes series
AbstractWe give criteria for the rationality of Cantor series and series where a1, a2, ··· and b1, b2, ··· are integers such that an > 0 and the series converge. We precisely say when is rational (i) if {an}n=1∞ is a monotonic sequence of integers and bn + 1 − bn = o(an + 1) or lim and (ii) if for all large n. We give similar criteria for the rationality of Ahmes series and more generally series . For example, if bn > 0 and lim , where An = lcm(a1, a2, ···, an), then is rational if and only if for large n.On the other hand, we show that such results are impossible without growth restrictions. For example, we show that for any integers d > c > 1 there is a sequence {bn}n = 1∞ such that every number x from some interval can be represented as with an ∈ {c,d} for all n
Multivariate Diophantine equations with many solutions
We show that for each n-tuple of positive rational integers (a_1,..,a_n)
there are sets of primes S of arbitrarily large cardinality s such that the
solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not
contained in fewer than exp((4+o(1))s^{1/2}(log s)^{-1/2}) proper linear
subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for
m=2 [Compositio 36 (1988), 37-56].
Furthermore we prove that for any algebraic number field K of degree n, any
integer m with 1<=m<n, and any sufficiently large s there are integers
b_0,...,b_m in a number field which are linearly independent over the
rationals, and prime numbers p_1,...,p_s, such that the norm polynomial
equation |N_{K/Q}(b_0+b_1x_1+...+b_mx_m)|=p_1^{z_1}...p_s^{z_s} has at least
exp{(1+o(1)){n/m}s^{m/n}(log s)^{-1+m/n}) solutions in integers
x_1,..,x_m,z_1,..,z_s. This generalizes a result of Moree and Stewart [Indag.
Math. 1 (1990), 465-472].
Our main tool, also established in this paper, is an effective lower bound
for the number of ideals in a number field K of norm <=X composed of prime
ideals which lie outside a given finite set of prime ideals T and which have
norm <=Y. This generalizes a result of Canfield, Erdos and Pomerance [J. Number
Th. 17 (1983), 1-28], and of Moree and Stewart (see above).Comment: 29 page
On conjectures and problems of Ruzsa concerning difference graphs of S-units
Given a finite nonempty set of primes S, we build a graph with
vertex set by connecting x and y if the prime divisors of both the
numerator and denominator of x-y are from S. In this paper we resolve two
conjectures posed by Ruzsa concerning the possible sizes of induced
nondegenerate cycles of , and also a problem of Ruzsa concerning
the existence of subgraphs of which are not induced subgraphs.Comment: 15 page
Convergence properties of linear recurrence sequences
Analysis and Stochastic
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