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Regular number
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<p>Regular numbers are numbers that evenly divide powers of <a href="/facts/60_(number)/EawDWoBh">60</a> (or, equivalently, powers of <a href="/facts/30_(number)/JrwluRu2">30</a>). Equivalently, they are the numbers whose only prime divisors are <a href="/facts/2/u5WC44t9">2</a>, <a href="/facts/3/6QzW9RkE">3</a>, and <a href="/facts/5/47v53K62">5</a>. As an example, 602 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular. </p><p>These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study. </p> <ul><li>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their <a href="/facts/Prime_factor/L20FLkgn">prime factors</a>. This is a specific case of the more general k-<a href="/facts/Smooth_number/pC6DVLJW">smooth numbers</a>, the numbers that have no prime factor greater than k.</li> <li>In the study of <a href="/facts/Babylonian_mathematics/9lN3d6ML">Babylonian mathematics</a>, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the <a href="/facts/Sexagesimal/bspNRWdR">sexagesimal</a> (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.</li> <li>In <a href="/facts/Music_theory/BoekvflP">music theory</a>, regular numbers occur in the ratios of tones in <a href="/facts/Five-limit_tuning/3R5QIJF3">five-limit</a> <a href="/facts/Just_intonation/TM91PueU">just intonation</a>. In connection with music theory and related theories of <a href="/facts/Architecture/FccThs8G">architecture</a>, these numbers have been called the harmonic whole numbers.</li> <li>In <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, regular numbers are often called Hamming numbers, after <a href="/facts/Richard_Hamming/plAXy9p0">Richard Hamming</a>, who proposed the problem of finding computer <a href="/facts/Algorithm/fnl5NmRt">algorithms</a> for generating these numbers in ascending order. This problem has been used as a test case for <a href="/facts/Functional_programming/cdxqPEIM">functional programming</a>.</li></ul>