We establish nontrivial bounds for the least common multiple (LCM) within arithmetic progressions. We successfully derive efficient bounds that are, to a certain extent, optimal for these progressions. In this paper, we focus on the integer <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{lcm}\{a,a + b,\ldots ,a + nb\},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>lcm</mtext> <mo stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>n</mi> <mi>b</mi> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ a,b,n\in \mathbb {N}^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gcd (a,b)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that there exists a divisor <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_{a,b,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and a multiple <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m_{a,b,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, both of which depend simply on a, b and n. Moreover, we prove that both equalities <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{lcm}\{a,a + b,\ldots ,a + nb\}=d_{a,b,n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>lcm</mtext> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>n</mi> <mi>b</mi> <mo stretchy="false">}</mo> </mrow> <mo>=</mo> <msub> <mi>d</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{lcm}\{a,a + b,\ldots ,a + nb\}=m_{a,b,n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>lcm</mtext> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>n</mi> <mi>b</mi> <mo stretchy="false">}</mo> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> hold for an infinite number of triples (a, b, n).

Nontrivial bounds for the least common multiple of finite arithmetic progressions

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发表日期：

2026年6月3日

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Hongguang Wu

摘要

We establish nontrivial bounds for the least common multiple (LCM) within arithmetic progressions. We successfully derive efficient bounds that are, to a certain extent, optimal for these progressions. In this paper, we focus on the integer $\textrm{lcm}\{a,a + b,\ldots ,a + nb\},$ $lcm {a, a + b, \dots, a + n b},$ where $ a,b,n\in \mathbb {N}^{*}$ $a, b, n \in N^{*}$ and $\gcd (a,b)=1$ $gcd (a, b) = 1$ . We show that there exists a divisor $d_{a,b,n}$ $d_{a, b, n}$ and a multiple $m_{a,b,n}$ $m_{a, b, n}$ , both of which depend simply on a, b and n. Moreover, we prove that both equalities $\textrm{lcm}\{a,a + b,\ldots ,a + nb\}=d_{a,b,n}$ $lcm {a, a + b, \dots, a + n b} = d_{a, b, n}$ and $\textrm{lcm}\{a,a + b,\ldots ,a + nb\}=m_{a,b,n}$ $lcm {a, a + b, \dots, a + n b} = m_{a, b, n}$ hold for an infinite number of triples (a, b, n).