In this study, we find all Fibonacci and Lucas numbers which can be expressible as a product of two repdigits in the base b. It is shown that the largest Fibonacci and Lucas numbers which can be expressible as a produ...
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In this study, we find all Fibonacci and Lucas numbers which can be expressible as a product of two repdigits in the base b. It is shown that the largest Fibonacci and Lucas numbers which can be expressible as a product of two repdigits are F-12 = 144 and L-15 = 1364, respectively. Also, we have the presentation F-12 = 144 = 6 x (3 + 3. 7) = (6)(7) x (33)(7) = 4 x (4 + 4.8) = (4)(8) x (44)(8) and L-15 = 1364 x (22222)(4) = 2 x (2 + 2.4 + 2. 4(2) + 2.4(3) + 2.4(4)).
Let r >= 1 be an integer and U := (U-n)(n >= 0) be the Lucas sequence given by U-0 = 0, U-1 = 1, and Un+2 = rU(n+1) + U-n, for all n >= 0. In this paper, we show that there are no positive integers r >= 3,...
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Let r >= 1 be an integer and U := (U-n)(n >= 0) be the Lucas sequence given by U-0 = 0, U-1 = 1, and Un+2 = rU(n+1) + U-n, for all n >= 0. In this paper, we show that there are no positive integers r >= 3, x not equal 2, n >= 1 such that U-n(x) + U-n+1(x) is a member of U.
In this paper, we explicitly find all the solutions of the exponential Diophantine equation F-n+1(x) + F-n(x) - F-n-1(x) = F-m, in nonnegative integers (m, n, x), where F-n is the nth Fibonacci number.
In this paper, we explicitly find all the solutions of the exponential Diophantine equation F-n+1(x) + F-n(x) - F-n-1(x) = F-m, in nonnegative integers (m, n, x), where F-n is the nth Fibonacci number.
In this paper, we provide some results on the arithmetic nature of numbers related to U-numbers. In particular, we show the transcendence of a(?), for all algebraic number a ?/ {0, 11 and all Liouville number ?.
In this paper, we provide some results on the arithmetic nature of numbers related to U-numbers. In particular, we show the transcendence of a(?), for all algebraic number a ?/ {0, 11 and all Liouville number ?.
In this paper, we find all Fibonacci numbers which are products of two Jacobsthal numbers. Also we find all Jacobsthal numbers which are products of two Fibonacci numbers. More generally, taking k, m, n as positive in...
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In this paper, we find all Fibonacci numbers which are products of two Jacobsthal numbers. Also we find all Jacobsthal numbers which are products of two Fibonacci numbers. More generally, taking k, m, n as positive integers, it is proved that F-k = J(m)J(n) implies that (k, m, n) =(1, 1, 1), (2, 1, 1), (1, 1, 2), (2, 1, 2), (1, 2, 2), (2, 2, 2), (4, 1, 3), (4, 2, 3), (5, 1, 4), (5, 2, 4), (10, 4, 5), (8, 1, 6), (8, 2, 6) and J(k) = FmFn implies that (k, m, n) = (1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1), (1, 2, 2), (2, 2, 2), (3, 4, 1), (3, 4, 2), (4, 5, 1), (4, 5, 2), (6, 8, 1), (6, 8, 2).
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.
Let k >= 2 and let (P-n((k)))(n >= 2-k) be the k-generalized Pell sequence defined by P-n((k)) = 2P(n-1)((k)) + P-n-2((k)) + ... + P-n-k((k)) for n >= 2 with initial conditions P--(k-2)((k)) = P--(k-3)((k)) =...
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Let k >= 2 and let (P-n((k)))(n >= 2-k) be the k-generalized Pell sequence defined by P-n((k)) = 2P(n-1)((k)) + P-n-2((k)) + ... + P-n-k((k)) for n >= 2 with initial conditions P--(k-2)((k)) = P--(k-3)((k)) = ... = P--1((k)) = P-0((k) )= 0, P-1((k)) = 1. In this paper, we show that 12,13, 29, 33, 34, 70,84, 88, 89, 228, and 233 are the only k-generalized Pell numbers, which are concatenation of two repdigits with at least two digits.
In this study, we show that if 2 < m < n and Fm Fn represents a repdigit, then (m, n) belongs to the set {(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)}. Also, we show that if 0 < m ...
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In this study, we show that if 2 < m < n and Fm Fn represents a repdigit, then (m, n) belongs to the set {(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)}. Also, we show that if 0 < m < n and Lm L represents a repdigit, then (m, n) belongs to the set (0, 0), (0, 1), (1, 1), (0, 2), (1, 2). (2, 2), (0, 3), (1, 3), (1, 4). (1, 5), (2,5), (3, 5), (4,5)}.
For an integer k >= 2, let {F-n((k))}(n >= 2-k) be the k-generalized Fibonacci sequence which starts with 0, ... ,0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms....
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For an integer k >= 2, let {F-n((k))}(n >= 2-k) be the k-generalized Fibonacci sequence which starts with 0, ... ,0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c with at least two representations as a difference between a k-generalized Fibonacci number and a power of 3. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.
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