Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Daniel M. Kane; Scott Duke Kominers

}
//]]>
</script>
<input type="hidden" name="__VIEWSTATEGENERATOR" id="__VIEWSTATEGENERATOR" value="E39AC016" />
<script type="text/javascript">RegisterSod("sp.core.js", "\u002f_layouts\u002fsp.core.js?rev=7ByNlH\u00252BvcgRJg\u00252BRCctdC0w\u00253D\u00253D");</script><script type="text/javascript">RegisterSod("sp.res.resx", "\u002f_layouts\u002fScriptResx.ashx?culture=en\u00252Dus\u0026name=SP\u00252ERes\u0026rev=b6\u00252FcRx1a6orhAQ\u00252FcF\u00252B0ytQ\u00253D\u00253D");</script><script type="text/javascript">RegisterSod("sp.ui.dialog.js", "\u002f_layouts\u002fsp.ui.dialog.js?rev=Tpcmo1\u00252FSu6R0yewHowDl5g\u00253D\u00253D");RegisterSodDep("sp.ui.dialog.js", "sp.core.js");RegisterSodDep("sp.ui.dialog.js", "sp.res.resx");</script><script type="text/javascript">RegisterSod("core.js", "\u002f_layouts\u002f1033\u002fcore.js?rev=RiGU6\u00252FvAzNgOjxKFQLw9pw\u00253D\u00253D");</script>
<script type="text/javascript">
//<![CDATA[
var _spFormDigestRefreshInterval = 1440000;window.g_updateFormDigestPageLoaded = new Date(); window.g_updateFormDigestPageLoaded.setDate(window.g_updateFormDigestPageLoaded.getDate() -5);function loadMDN2() { EnsureScript('MDN.js', typeof(loadFilterFn), null); }
function loadMDN1() { ExecuteOrDelayUntilScriptLoaded(loadMDN2, 'sp.ribbon.js'); }
_spBodyOnLoadFunctionNames.push('loadMDN1');//]]>
</script>
<script type="text/javascript">RegisterSod("sp.runtime.js", "\u002f_layouts\u002fsp.runtime.js?rev=IGffcZfunndj0247nOxKVg\u00253D\u00253D");RegisterSodDep("sp.runtime.js", "sp.core.js");RegisterSodDep("sp.runtime.js", "sp.res.resx");</script><script type="text/javascript">RegisterSod("sp.js", "\u002f_layouts\u002fsp.js?rev=\u00252B4ZEyA892P3T0504qi0paw\u00253D\u00253D");RegisterSodDep("sp.js", "sp.core.js");RegisterSodDep("sp.js", "sp.runtime.js");RegisterSodDep("sp.js", "sp.ui.dialog.js");RegisterSodDep("sp.js", "sp.res.resx");</script><script type="text/javascript">RegisterSod("cui.js", "\u002f_layouts\u002fcui.js?rev=OOyJv78CADNBeet\u00252FvTvniQ\u00253D\u00253D");</script><script type="text/javascript">RegisterSod("inplview", "\u002f_layouts\u002finplview.js?rev=WB6Gy8a027aeNCq7koVCUg\u00253D\u00253D");RegisterSodDep("inplview", "core.js");RegisterSodDep("inplview", "sp.js");</script><script type="text/javascript">RegisterSod("ribbon", "\u002f_layouts\u002fsp.ribbon.js?rev=F\u00252BUEJ66rbXzSvpf7nN69wQ\u00253D\u00253D");RegisterSodDep("ribbon", "core.js");RegisterSodDep("ribbon", "sp.core.js");RegisterSodDep("ribbon", "sp.js");RegisterSodDep("ribbon", "cui.js");RegisterSodDep("ribbon", "sp.res.resx");RegisterSodDep("ribbon", "sp.runtime.js");RegisterSodDep("ribbon", "inplview");</script><script type="text/javascript">RegisterSod("sp.ui.policy.resources.resx", "\u002f_layouts\u002fScriptResx.ashx?culture=en\u00252Dus\u0026name=SP\u00252EUI\u00252EPolicy\u00252EResources\u0026rev=YhBHGmUAGyJ3lAgSdE4V\u00252Fw\u00253D\u00253D");</script><script type="text/javascript">RegisterSod("mdn.js", "\u002f_layouts\u002fmdn.js?rev=gwmFFJ2\u00252FfFacqXWAqG\u00252FqKg\u00253D\u00253D");RegisterSodDep("mdn.js", "sp.core.js");RegisterSodDep("mdn.js", "sp.runtime.js");RegisterSodDep("mdn.js", "sp.js");RegisterSodDep("mdn.js", "cui.js");RegisterSodDep("mdn.js", "ribbon");RegisterSodDep("mdn.js", "sp.ui.policy.resources.resx");</script>
<script type="text/javascript">
//<![CDATA[

function _spNavigateHierarchyEx(nodeDiv, dataSourceId, dataPath, url, listInContext, type, additionalQString) {
    SetAdditionalNavigateHierarchyQString(additionalQString);
    g_originalSPNavigateFunc(nodeDiv, dataSourceId, dataPath, url, listInContext, type);
}

g_originalSPNavigateFunc = _spNavigateHierarchy;
_spNavigateHierarchy = _spNavigateHierarchyEx;
var _fV4UI = true;//]]>
</script>
<script type="text/javascript">RegisterSod("msstring.js", "\u002f_layouts\u002f1033\u002fmsstring.js?rev=QtiIcPH3HV7LgVSO7vONFg\u00253D\u00253D");</script><script type="text/javascript">RegisterSod("browserScript", "\u002f_layouts\u002f1033\u002fnon_ie.js?rev=EVTj1bu32\u00252FMla6SDN\u00252FsNTA\u00253D\u00253D");</script>
<script type="text/javascript">
//<![CDATA[
var __wpmExportWarning='This Web Part Page has been personalized. As a result, one or more Web Part properties may contain confidential information. Make sure the properties contain information that is safe for others to read. After exporting this Web Part, view properties in the Web Part description file (.WebPart) by using a text editor such as Microsoft Notepad.';var __wpmCloseProviderWarning='You are about to close this Web Part.  It is currently providing data to other Web Parts, and these connections will be deleted if this Web Part is closed.  To close this Web Part, click OK.  To keep this Web Part, click Cancel.';var __wpmDeleteWarning='You are about to permanently delete this Web Part.  Are you sure you want to do this?  To delete this Web Part, click OK.  To keep this Web Part, click Cancel.';//]]>
</script>
</form>

Go to main content

Article | Canadian Mathematical Bulletin

Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

by Daniel M. Kane and Scott Duke Kominers

Abstract

For relatively prime positive integers and r, we consider the least common multiple L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n) of the finite arithmetic progression \{u_k:=u_0+kr\}_{k=0}^n. We derive new lower bounds on L_n that improve upon those obtained previously when either u_0 or n is large. When r is prime, our best bound is sharp up to a factor of n+1 for u_0 properly chosen, and is also nearly sharp as n\to\infty.

Keywords: Mathematical Methods;

Format: Print

Find at Harvard Purchase