In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an S-unit whose coefficients are bounded. In particular, we fin...
详细信息
In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an S-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an S-unit associated to the set of primes {2, 3, 5, 7}.
In this paper, we show that 204 and 1189 are the only balancing numbers which are concatenation of three repdigits and that 3363 is the only Lucas-balancing number of this form.
In this paper, we show that 204 and 1189 are the only balancing numbers which are concatenation of three repdigits and that 3363 is the only Lucas-balancing number of this form.
In this paper, we determine all repdigits in base b for 2 = 5 and n >= 1;respectively, where 1 <= m <= n.
In this paper, we determine all repdigits in base b for 2 <= b <= 10;which are products of two Pell numbers or Pell-Lucas numbers. It is shown that the largest Pell number which is a base b-repdigit is P-6 = 70 = (77)(9) = 7 + 7.9. Also, we give the result that the equations PmPn + 1 = b(k) and Q(m)Q(n) + 1 = b(k) have no solutions for n >= 5 and n >= 1;respectively, where 1 <= m <= n.
Padovan and Perrin sequences are ternary recurrent sequences that satisfy the same relation w(n) = w(n-2 )+ w(n-3) with different initial conditions (w(0), w(1), w(2)) = (1, 1, 1) and (3, 0, 2), respectively. In this ...
详细信息
Padovan and Perrin sequences are ternary recurrent sequences that satisfy the same relation w(n) = w(n-2 )+ w(n-3) with different initial conditions (w(0), w(1), w(2)) = (1, 1, 1) and (3, 0, 2), respectively. In this study we compute all pairs of Padovan and Perrin numbers that are multiplicatively dependent.
In this paper, we consider the D(+/- k)-triple {k -/+ 1, k, 4k -/+ 1} and we prove that, if k is not a perfect square then: (1) There is no d such that {k-1, k, 4k-1, d} is a D(k)-quadruple;(2) If {k, k + 1, 4k + 1, d...
详细信息
In this paper, we consider the D(+/- k)-triple {k -/+ 1, k, 4k -/+ 1} and we prove that, if k is not a perfect square then: (1) There is no d such that {k-1, k, 4k-1, d} is a D(k)-quadruple;(2) If {k, k + 1, 4k + 1, d} is a D(-k)-quadruple, then d = 1. This extends a work done by Fujita [13].
A positive integer n is said to be a perfect number, if sigma-(n) = 2n, where sigma (N) is the sum of all positive divisors of N. Luca (Rend Circ Mat Palermo Ser 49:313-318, 2000) proved that there is no perfect numbe...
详细信息
A positive integer n is said to be a perfect number, if sigma-(n) = 2n, where sigma (N) is the sum of all positive divisors of N. Luca (Rend Circ Mat Palermo Ser 49:313-318, 2000) proved that there is no perfect number in the Fibonacci sequence. For k >= 2, the k-generalized Fibonacci sequence (F-n((k)))(n) is defined by the initial values 0, 0, ... 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we prove, among other things, that there is no even perfect numbers belonging to k-generalized Fibonacci sequences when k not equivalent to 3 (mod 4).
In this paper, we give an algorithm which finds, for an integer base b >= 2, all squarefree integers d >= 2 such that sequence of X-components {X-n}(n >= 1) of the Pell equation X-2-dY(2) = +/- 1 has two memb...
详细信息
In this paper, we give an algorithm which finds, for an integer base b >= 2, all squarefree integers d >= 2 such that sequence of X-components {X-n}(n >= 1) of the Pell equation X-2-dY(2) = +/- 1 has two members which are baseb-repdigits. We implement this algorithm and find all the solutions to this problem for all bases b is an element of[2, 100].
We show that detecting real roots for honestly n-variate (n + 2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bo...
详细信息
ISBN:
(纸本)9781605586090
We show that detecting real roots for honestly n-variate (n + 2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a characterization of those functions k(n) such that the complexity of detecting real roots for n-variate (n + k(n))-nomials transitions from P to NP-hardness as n -> infinity. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of A-discriminants of n-variate(n+k(n))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.
暂无评论