<abstract><p>In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p > 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> or of repunits <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p > 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.</p></abstract>
In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let a 1 , a 2 , . . . , a l be positive integers such that their greatest common divisor is one. For a nonnegative integer p, denote the p-Frobenius number by g p (a 1 , a 2 , . . . , a l ), which is the largest integer that can be represented at most p ways by a linear combination with nonnegative integer coefficients of a 1 , a 2 , . . . , a l . When p = 0, 0-Frobenius number is the classical Frobenius number. When l = 2, p-Frobenius number is explicitly given. However, when l = 3 and even larger, even in special cases, it is not easy to give the Frobenius number explicitly, and it is even more difficult when p > 0, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers or of repunits for the case where l = 3. In this paper, we show the explicit formula for the Fibonacci triple when p > 0. In addition, we give an explicit formula for the p-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most p ways. Furthermore, explicit formulas are shown concerning the Lucas triple.
For given positive integers a 1 , a 2 , . . . , a k with gcd(a 1 , a 2 , . . . , a k ) = 1, consider the number of nonnegative solutions (x 1 , x 2 , . . . , x k ) of the linear equation a 1 x 1 + a 2 x 2 + • • • + a k x k = n for a positive integer n. For a given nonnegative integer p, there is a maximum n such that the number of nonnegative integer solutions is at most p, and it is very attractive to find the explicit formula of the maximum n. In fact, when p = 0, such a problem of finding the maximum integer n is called the linear Diophantine problem of Frobenius, and this maximum n is called the Frobenius number. The explicit formula for two variables is known not only for p = 0 but also for p > 0, but when there are three or more variables, it is difficult even in the special case of p = 0. For p > 0, it is not only more difficult, but no explicit formula has been found. In this paper, an explicit formula of such a generalized Frobenius number is given for the sequence of arithmetic progressions, in particular, for three variables. Further, explicit formulas are given not only for the maximum value of n such that the number of solutions is at most p, but also for the total number, the sum, and the weighted sum of such solutions.
The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation a 1 x 1 + • • • + a k x k = n (a 1 , . . . , a k are given positive integers with gcd(a 1 , . . . , a k ) = 1) does not have a non-negative integer solution (x 1 , . . . , x k ). The generalized Frobenius number (called the p-Frobenius number) is the largest integer such that this linear equation has at most p solutions. That is, when p = 0, the 0-Frobenius number is the original Frobenius number.In this paper, we introduce and discuss p-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer p, p-gaps, p-symmetric semigroups, p-pseudosymmetric semigroups, and the like are defined, and their properties are obtained. When p = 0, they correspond to the original gaps, symmetric semigroups, and pseudo-symmetric semigroups, respectively.
The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.
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