It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

Given a pair (Ut) and (Vt) of Lucas sequences, we establish various congruences involving sums of ratios Vt Ut . More precisely, let p be a prime divisor of the positive integer m. We establish congruences, modulo powers of p, for the sum ∑ Vt Ut , where t runs from 1 to r(m), the rank of m, and r(q) ∤ t for all prime factors q of m.

Let G be a finite abelian group with |G| > 1. Let a1, . . . , ak be k distinct elements of G and let b1, . . . , bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the smallest prime divisor of |G|. We show that there is a permutation π on {1, . . . , k} such that a1bπ(1), . . . , akbπ(k) are distinct, provided that any other prime divisor of |G| (if ther...

Let G be a finite abelian group with |G| > 1. Let a1, . . . , ak be k distinct elements of G and let b1, . . . , bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the smallest prime divisor of |G|. We show that there is a permutation π on {1, . . . , k} such that a1bπ(1), . . . , akbπ(k) are distinct, provided that any other prime divisor of |G| (if ther...

Let G be a finite abelian group with |G| > 1. Let a1, . . . , ak be k distinct elements of G and let b1, . . . , bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation π on {1, . . . , k} such that a1bπ(1), . . . , akbπ(k) are distinct, provided that any other prime divisor of |G| (if there i...

We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence {xn}∞ n=1 defined by the linear recurrence xn+1 = xn + xn−1 and the initial values x0 = 3 and x1 = 1. This is the first example of a ‘non-torsion’ second order recurrent sequence with irreducible recurrence relation for whi...

This paper provides further results on the perfect state transfer in integral circulant graphs (ICG graphs). The non-existence of PST is proved for several classes of ICG graphs containing an isolated divisor d0, i.e. the divisor which is relatively prime to all other divisors from d ∈ D \ {d0}. The same result is obtained for classes of integral circulant graphs having the NSF property (i.e. e...

from which it follows that dl(m) is a rational number with only a power of 2 in its denominator. Extensive calculations have shown that, with rare exceptions, the numerators of dl(m) contain a single large prime divisor and its remaining factors are very small. For example d6(30) = 2 12 · 7 · 11 · 13 · 17 · 31 · 37 · 639324594880985776531. Similarly, d10(200) has 197 digits with a prime factor ...

Let n > 1 be a positive integer and p be the smallest prime divisor of n. Let S be a sequence of elements from Zn = Z/nZ of length n + k where k ≥ np − 1. If every element of Zn appears in S at most k times, we prove that there must be a subsequence of S of length n whose sum is zero when n has only two distinct prime divisors.