The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908949857435648 |
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| author | Liu, Feihu Xin, Guoce Ye, Suting Yin, Jingjing |
| author_facet | Liu, Feihu Xin, Guoce Ye, Suting Yin, Jingjing |
| contents | Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1)$, $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$. Our idea also applies to the case $B=(b_1,b_2,...,b_k)$, $b_1> 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_09039 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$ Liu, Feihu Xin, Guoce Ye, Suting Yin, Jingjing Combinatorics Number Theory Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1)$, $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$. Our idea also applies to the case $B=(b_1,b_2,...,b_k)$, $b_1> 1$. |
| title | The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$ |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2304.09039 |