{"id":1396,"date":"2017-04-30T10:05:25","date_gmt":"2017-04-30T16:05:25","guid":{"rendered":"https:\/\/michaelhoke.name\/oldblog\/?p=1396"},"modified":"2026-06-07T13:03:04","modified_gmt":"2026-06-07T19:03:04","slug":"i-did-some-math-average-prime-powers-among-counting-numbers","status":"publish","type":"post","link":"https:\/\/michaelhoke.name\/oldblog\/2017\/04\/30\/i-did-some-math-average-prime-powers-among-counting-numbers\/","title":{"rendered":"I did some math: average prime powers among counting numbers"},"content":{"rendered":"\r\n<p>In the past few days, I&rsquo;ve seen a couple of good little math problems posted on Twitter. I follow a number of mathematicians, including <a href=\"http:\/\/www.jamestanton.com\/\">James Tanton<\/a> (<a href=\"https:\/\/twitter.com\/jamestanton\">@jamestanton<\/a>), who pretty regularly posts math problems that are sometimes easy enough that I have a fair chance of solving them (though he almost never provides any solutions, so it&rsquo;s sometimes hard to tell if I&rsquo;ve gotten it right). A couple of days ago, he asked a simple question:<\/p>\r\n<div class=\"embed-x\"><blockquote class=\"twitter-tweet\" data-width=\"500\" data-dnt=\"true\"><p lang=\"en\" dir=\"ltr\">What is the average power of a given prime p among the prime factorizations of the counting numbers?<\/p>&mdash; James Tanton (@jamestanton) <a href=\"https:\/\/x.com\/jamestanton\/status\/857214222798737408?ref_src=twsrc%5Etfw\">April 26, 2017<\/a><\/blockquote><script async src=\"https:\/\/platform.x.com\/widgets.js\" charset=\"utf-8\"><\/script><\/div>\r\n<p>This is not a terribly difficult problem to solve as long as you can sum a particular doubly-infinite series, but summing that series is probably the most interesting part of the problem.<\/p>\r\n<!--more-->\r\n<p>I&rsquo;d like to think I could have figured it out on my own, but I&rsquo;ll never know, because before I tried to solve the problem, I watched <a href=\"https:\/\/mikesmathpage.wordpress.com\/2017\/04\/27\/a-nice-problem-about-primes-for-kids-from-james-tanton\/\">the discussion another mathematician had with his two sons<\/a> about the problem during their family math night. I love the idea of having family math night discussions, and I am particularly impressed with his children&rsquo;s willingness to give some real thought to tackling this problem. His sons can&rsquo;t be very old (they don&rsquo;t look old enough to me to be in high school yet), and yet one of them immediately had the critical insight of how to sum the series. I have a feeling that without his suggestion, I might have struggled for awhile before coming up with the solution myself.<\/p>\r\n<p>The question asks for the &ldquo;average&rdquo; power of a given prime <i>p<\/i> among the counting numbers. A prime <i>p<\/i> is a factor in every <i>p<\/i>th counting number&#8212;which means that it is <em>not<\/em> a factor in <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> of the counting numbers, in which the power of <i>p<\/i> is 0 (there's a subtlety here about measuring proportions of the infinite set of counting numbers, but think of looking at the proportion of the first <i>n<\/i> counting numbers that have a factor <i>p<\/i>, then taking the limit as <i>n<\/i> approaches infinity). Of the 1\/<i>p<\/i> counting numbers that have <i>p<\/i> as a factor, you can divide by <i>p<\/i> and recover the counting numbers. So <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> of these numbers have no further <i>p<\/i> factors, and <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> of the counting numbers have a power of <i>p<\/i> equal to 1. Divide the remaining numbers again by <i>p<\/i>, and the argument continues: <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>2<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> of the counting numbers have a power of <i>p<\/i> equal to 2, <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>3<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> have a power of <i>p<\/i> equal to 3, and <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>n<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> have a power of <i>p<\/i> equal to <i>n<\/i>. Muliplying the proportions by all of the possible powers (all of the counting numbers) yields the average power of <i>p<\/i> among all of the counting numbers:<\/p>\r\n\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mtext>Average power of<\/mtext>\r\n  <mspace width=\"0.5em\" \/>\r\n  <mi>p<\/mi>\r\n  <mo>=<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mo>(<\/mo>\r\n      <mi>p<\/mi>\r\n      <mo>-<\/mo>\r\n      <mn>1<\/mn>\r\n      <mo>)<\/mo>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <munderover>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>n<\/mi>\r\n        <mo>=<\/mo>\r\n        <mn>1<\/mn>\r\n      <\/mrow>\r\n        <mo>&infin;<\/mo>\r\n  <\/munderover>\r\n  <mrow>\r\n    <mo>(<\/mo>\r\n    <mi>n<\/mi>\r\n    <mo>&amp;CenterDot;<\/mo>\r\n    <mfrac>\r\n      <mrow>\r\n        <mn>1<\/mn>\r\n      <\/mrow>\r\n      <mrow>\r\n        <msup>\r\n          <mi>p<\/mi>\r\n          <mi>n<\/mi>\r\n        <\/msup>\r\n      <\/mrow>\r\n    <\/mfrac>\r\n    <mo>)<\/mo>\r\n  <\/mrow>\r\n  <\/mfenced>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>It&rsquo;s easy enough to sum just the geometric series <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>n<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, but in this series, the terms are scaled by <i>n<\/i>: so there is one term <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, two <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>2<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> terms, three <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>3<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> terms, etc. The challenge is to see that this means there are an infinite number of ordinary geometric sequences: one that starts with <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, then another starting at <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>2<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, and another starting at <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>3<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, and so on. In the third video on the family-math-night page linked above, the older son comes up with a brilliant way of seeing this structure&#8212;by re-writing the series with each power on its own line, and then grouping terms vertically:<\/p>\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <munderover>\r\n    <mo>&amp;Sum;<\/mo>\r\n    <mrow>\r\n      <mi>n<\/mi>\r\n      <mo>=<\/mo>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mo>&infin;<\/mo>\r\n  <\/munderover>\r\n  <mrow>\r\n    <mo>(<\/mo>\r\n    <mi>n<\/mi>\r\n    <mo>&amp;CenterDot;<\/mo>\r\n    <mfrac>\r\n      <mrow>\r\n        <mn>1<\/mn>\r\n      <\/mrow>\r\n      <mrow>\r\n        <msup>\r\n          <mi>p<\/mi>\r\n          <mi>n<\/mi>\r\n        <\/msup>\r\n      <\/mrow>\r\n    <\/mfrac>\r\n    <mo>)<\/mo>\r\n  <\/mrow>\r\n  <mo>=<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>+<\/mo>\r\n  <mn>2<\/mn>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>2<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>+<\/mo>\r\n  <mn>3<\/mn>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>3<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>+<\/mo>\r\n  <mn>4<\/mn>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>4<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>+<\/mo>\r\n  <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mtable>\r\n    <mtr>\r\n      <mtd>\r\n        <mo>=<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <mi>p<\/mi>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n    <\/mtr>\r\n    <mtr>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>2<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>2<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n    <\/mtr>\r\n    <mtr>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>3<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>3<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>3<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n      <\/mtd>\r\n    <\/mtr>\r\n    <mtr>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>4<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>4<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>4<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n      <\/mtd>\r\n      <mtd>\r\n        <mo>+<\/mo>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <msup>\r\n                    <mi>p<\/mi>\r\n                  <mn>4<\/mn>\r\n            <\/msup>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n        <mo>+<\/mo>\r\n        <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n      <\/mtd>\r\n    <\/mtr>\r\n  <\/mtable>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mo>=<\/mo>\r\n  <mo>(<\/mo>\r\n  <mrow>\r\n    <mtable>\r\n      <mtr>\r\n        <mtd>\r\n          <mspace width=\"0.5em\" \/>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <mi>p<\/mi>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n        <\/mtd>\r\n      <\/mtr>\r\n      <mtr>\r\n        <mtd>\r\n          <mo>+<\/mo>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <msup>\r\n                <mi>p<\/mi>\r\n                <mn>2<\/mn>\r\n              <\/msup>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n        <\/mtd>\r\n      <\/mtr>\r\n      <mtr>\r\n        <mtd>\r\n          <mo>+<\/mo>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <msup>\r\n                <mi>p<\/mi>\r\n                <mn>3<\/mn>\r\n              <\/msup>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n        <\/mtd>\r\n      <\/mtr>\r\n      <mtr>\r\n        <mtd>\r\n          <mo>+<\/mo>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <msup>\r\n                <mi>p<\/mi>\r\n                <mn>4<\/mn>\r\n              <\/msup>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n        <\/mtd>\r\n      <\/mtr>\r\n      <mtr>\r\n        <mtd>\r\n          <mo>+<\/mo>\r\n          <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n        <\/mtd>\r\n      <\/mtr>\r\n    <\/mtable>\r\n  <\/mrow>\r\n  <mo>)<\/mo>\r\n  <mo>+<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mtable>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>2<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>3<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>4<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n          <\/mtd>\r\n        <\/mtr>\r\n      <\/mtable>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>+<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mtable>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>3<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>4<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n          <\/mtd>\r\n        <\/mtr>\r\n      <\/mtable>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>+<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mtable>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mspace width=\"0.5em\" \/>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <msup>\r\n                        <mi>p<\/mi>\r\n                      <mn>4<\/mn>\r\n                <\/msup>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n          <\/mtd>\r\n        <\/mtr>\r\n        <mtr>\r\n          <mtd>\r\n            <mo>+<\/mo>\r\n            <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n          <\/mtd>\r\n        <\/mtr>\r\n      <\/mtable>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>+<\/mo>\r\n  <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;<\/mo>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>This kind of insight is beautiful&#8212;and it is, at bottom, why I enjoy working through math problems. Most of the time I just struggle, but every now and then, my brain reorients and I find the blade to cut through the Gordian knot. It&rsquo;s a wonderful feeling.<\/p>\r\n\r\n<p>So (ignoring for a moment the <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> factor in front of the summation) we have a series of series: the first, which starts with <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, sums to <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->. The second is the same series as the first, with all of the terms multiplied by <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->. The third is the same as the first multiplied by <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>2<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->. The fourth is the first multiplied by <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mn>3<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->, etc. So the series of geometric series sums to&hellip; a geometric series:<\/p>\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <munderover>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>n<\/mi>\r\n        <mo>=<\/mo>\r\n        <mn>0<\/mn>\r\n      <\/mrow>\r\n        <mo>&infin;<\/mo>\r\n  <\/munderover>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mi>n<\/mi>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n<p>&hellip; with the sum taken over the <b>whole<\/b> numbers <i>n<\/i> (starting with 0, because the first term is 1, once the <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> is factored out), which is equal to <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n            <mrow>\r\n              <mfenced open=\"(\" close=\")\" separators=\",\">\r\n                <mrow>\r\n                  <mi>p<\/mi>\r\n                  <mo>-<\/mo>\r\n                  <mn>1<\/mn>\r\n                <\/mrow>\r\n              <\/mfenced>\r\n            <\/mrow>\r\n            <mn>2<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw -->. But we previously had a factor of <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <mfrac>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> that we ignored. Putting that back in, we have:<\/p>\r\n\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mtext>Average power of<\/mtext>\r\n  <mspace width=\"0.5em\" \/>\r\n  <mi>p<\/mi>\r\n  <mo>=<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&amp;CenterDot;<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n    <mrow>\r\n      <msup>\r\n            <mrow>\r\n              <mfenced open=\"(\" close=\")\" separators=\",\">\r\n                <mrow>\r\n                  <mi>p<\/mi>\r\n                  <mo>-<\/mo>\r\n                  <mn>1<\/mn>\r\n                <\/mrow>\r\n              <\/mfenced>\r\n            <\/mrow>\r\n            <mn>2<\/mn>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>=<\/mo>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>This is a nice little result in itself: among the counting numbers, larger primes occur less frequently, of course, but they appear regularly&#8212;and one expects each to have, on average, a factor of <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <msup>\r\n          <mi>p<\/mi>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <mfenced open=\"(\" close=\")\" separators=\",\">\r\n              <mrow>\r\n                <mi>p<\/mi>\r\n                <mo>-<\/mo>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n            <\/mfenced>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n  <\/msup>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> in any given counting number. A few days after posting this question, though, Tanton asked a follow-up question:<\/p>\r\n<div class=\"embed-x\"><blockquote class=\"twitter-tweet\" data-width=\"500\" data-dnt=\"true\"><p lang=\"en\" dir=\"ltr\">From yesterday, 2^(1)*3^(1\/2)*5^(1\/4)*7^(1\/6)*11^(1\/10)*13^(1\/12)*... is the most average prime factorization. Is this number finite or inf?<\/p>&mdash; James Tanton (@jamestanton) <a href=\"https:\/\/x.com\/jamestanton\/status\/857587154674884608?ref_src=twsrc%5Etfw\">April 27, 2017<\/a><\/blockquote><script async src=\"https:\/\/platform.x.com\/widgets.js\" charset=\"utf-8\"><\/script><\/div>\r\n<p>Consider the product of <!-- raw -->\r\n<math>\r\n<mrow>\r\n  <msup>\r\n          <mi>p<\/mi>\r\n        <mfrac>\r\n          <mrow>\r\n            <mn>1<\/mn>\r\n          <\/mrow>\r\n          <mrow>\r\n            <mfenced open=\"(\" close=\")\" separators=\",\">\r\n              <mrow>\r\n                <mi>p<\/mi>\r\n                <mo>-<\/mo>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n            <\/mfenced>\r\n          <\/mrow>\r\n        <\/mfrac>\r\n  <\/msup>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --> over all of the primes&#8212;that is, construct the number that has the &ldquo;average&rdquo; power of every prime. Is this construction finite or infinite?<\/p>\r\n\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mi>P<\/mi>\r\n  <mo>=<\/mo>\r\n  <munder>\r\n          <mo>&amp;Product;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <msup>\r\n              <mi>p<\/mi>\r\n            <mfrac>\r\n              <mrow>\r\n                <mn>1<\/mn>\r\n              <\/mrow>\r\n              <mrow>\r\n                <mfenced open=\"(\" close=\")\" separators=\",\">\r\n                  <mrow>\r\n                    <mi>p<\/mi>\r\n                    <mo>-<\/mo>\r\n                    <mn>1<\/mn>\r\n                  <\/mrow>\r\n                <\/mfenced>\r\n              <\/mrow>\r\n            <\/mfrac>\r\n      <\/msup>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mtext>. Is<\/mtext>\r\n  <mspace width=\"0.5em\" \/>\r\n  <mi>P<\/mi>\r\n  <mspace width=\"0.5em\" \/>\r\n  <mtext>finite or infinite?<\/mtext>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>The answer should be obvious&#8212;the &ldquo;average&rdquo; counting number had better not be finite&#8212;but the fun is in proving it, and in this case, the solution may depend on a rather interesting result of Euler (as so many problems often do). But this result of Euler&rsquo;s is interesting in part because his &ldquo;proof&rdquo; was rather questionable. To add to the intrigue, Paul Erd\u0151s also gave a separate proof of the same result.<\/p>\r\n\r\n<p>Here&rsquo;s how I approached the problem. I don&rsquo;t like dealing with infinite products&#8212;the way my brain is wired, I find it much easier to deal with sums (though I imagine many real mathematicians wouldn&rsquo;t trouble themselves much about the difference). So I took the logarithm of the product:<\/p>\r\n\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mi>P<\/mi>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>=<\/mo>\r\n  <munder>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      \r\n        <mrow>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <mfenced open=\"(\" close=\")\" separators=\",\">\r\n                <mrow>\r\n                  <mi>p<\/mi>\r\n                  <mo>-<\/mo>\r\n                  <mn>1<\/mn>\r\n                <\/mrow>\r\n              <\/mfenced>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n          <mo>&amp;CenterDot;<\/mo>\r\n          <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n          <mfenced open=\"(\" close=\")\" separators=\",\">\r\n            <mrow>\r\n              <mi>p<\/mi>\r\n            <\/mrow>\r\n          <\/mfenced>\r\n        <\/mrow>\r\n      \r\n    <\/mrow>\r\n  <\/mfenced>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>I don&rsquo;t like logarithms much, either, so I want get rid of them if possible. Because I think the product is infinite, I&rsquo;m going to get rid of ln(<i>p<\/i>) by considering a lower bound. For all primes <i>p<\/i> &gt; 2, ln(<i>p<\/i>) &gt; ln(2), so:<\/p>\r\n\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mi>P<\/mi>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>=<\/mo>\r\n  <munder>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      \r\n        <mrow>\r\n          <mfrac>\r\n            <mrow>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n            <mrow>\r\n              <mfenced open=\"(\" close=\")\" separators=\",\">\r\n                <mrow>\r\n                  <mi>p<\/mi>\r\n                  <mo>-<\/mo>\r\n                  <mn>1<\/mn>\r\n                <\/mrow>\r\n              <\/mfenced>\r\n            <\/mrow>\r\n          <\/mfrac>\r\n          <mo>&amp;CenterDot;<\/mo>\r\n          <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n          <mfenced open=\"(\" close=\")\" separators=\",\">\r\n            <mrow>\r\n              <mi>p<\/mi>\r\n            <\/mrow>\r\n          <\/mfenced>\r\n        <\/mrow>\r\n      \r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <mo>&gt;<\/mo>\r\n  <munder>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mfrac>\r\n        <mrow>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n        <mrow>\r\n          <mfenced open=\"(\" close=\")\" separators=\",\">\r\n            <mrow>\r\n              <mi>p<\/mi>\r\n              <mo>-<\/mo>\r\n              <mn>1<\/mn>\r\n            <\/mrow>\r\n          <\/mfenced>\r\n        <\/mrow>\r\n      <\/mfrac>\r\n      <mo>&amp;CenterDot;<\/mo>\r\n      <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mn>2<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n<p><!-- raw -->\r\n<math display=\"block\">\r\n<mrow>\r\n  <mo>=<\/mo>\r\n  <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mn>2<\/mn>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <munder>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mfenced open=\"(\" close=\")\" separators=\",\">\r\n        <mrow>\r\n          <mi>p<\/mi>\r\n          <mo>-<\/mo>\r\n          <mn>1<\/mn>\r\n        <\/mrow>\r\n      <\/mfenced>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n  <mo>&gt;<\/mo>\r\n  <mi>ln<\/mi><mo>&amp;ApplyFunction;<\/mo>\r\n  <mfenced open=\"(\" close=\")\" separators=\",\">\r\n    <mrow>\r\n      <mn>2<\/mn>\r\n    <\/mrow>\r\n  <\/mfenced>\r\n  <munder>\r\n          <mo>&amp;Sum;<\/mo>\r\n      <mrow>\r\n        <mi>p<\/mi>\r\n        <mo>&isin;<\/mo>\r\n        <mtext>primes<\/mtext>\r\n      <\/mrow>\r\n  <\/munder>\r\n  <mfrac>\r\n    <mrow>\r\n      <mn>1<\/mn>\r\n    <\/mrow>\r\n    <mrow>\r\n      <mi>p<\/mi>\r\n    <\/mrow>\r\n  <\/mfrac>\r\n<\/mrow>\r\n<\/math>\r\n<!-- \/raw --><\/p>\r\n\r\n<p>If the right-hand side is infinite, then so is the left-hand side. The question is whether the sum of the reciprocals of the primes diverges. I know that the harmonic series diverges (slowly!), but the prime reciprocals is a rather sparse subsequence of the harmonics. If it diverges, it must do so incredibly slowly. But diverge it does, so the product considered above is infinite.<\/p>\r\n\r\n<p>According to Wikipedia, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Divergence_of_the_sum_of_the_reciprocals_of_the_primes#Euler.27s_proof\">Euler gave a questionable proof that the sum of the prime reciprocals does, in fact, diverge<\/a>. But I like <a href=\"https:\/\/en.wikipedia.org\/wiki\/Divergence_of_the_sum_of_the_reciprocals_of_the_primes#Erd.C5.91s.27s_proof_by_upper_and_lower_estimates\">Erd\u0151s&rsquo;s proof by contradiction given on the same page<\/a>, which considers the number of counting numbers less than or equal to <i>x<\/i> that are products of the first <i>k<\/i> primes. Erd\u0151s forms both lower and upper bounds, and shows that if the sum of prime reciprocals converges those two bounds contradict each other. It is not a difficult proof to follow, but it is hardly an obvious proof&#8212;there is real brilliance in the set-up, and I could never have come up with it myself.<\/p>\r\n<p>Working through problems like these really makes me want to attend the math camp for adults that Tanton posted about on Friday:<\/p>\r\n<div class=\"embed-x\"><blockquote class=\"twitter-tweet\" data-width=\"500\" data-dnt=\"true\"><p lang=\"en\" dir=\"ltr\">Math Camp for adults! <a href=\"https:\/\/t.co\/tjmwuhfGZD\">https:\/\/t.co\/tjmwuhfGZD<\/a> Woohoo!<\/p>&mdash; James Tanton (@jamestanton) <a href=\"https:\/\/x.com\/jamestanton\/status\/858163162201419776?ref_src=twsrc%5Etfw\">April 29, 2017<\/a><\/blockquote><script async src=\"https:\/\/platform.x.com\/widgets.js\" charset=\"utf-8\"><\/script><\/div>\r\n<p>I probably can&rsquo;t make it, since I don&rsquo;t have much vacation at work, but I sure wish I could!<\/p>","protected":false},"excerpt":{"rendered":"In the past few days, I&rsquo;ve seen a couple of good little math problems posted on Twitter. I follow a number of mathematicians, including James Tanton (@jamestanton), who pretty regularly posts math problems that are sometimes easy enough that I have a fair chance of solving them (though he almost never provides any solutions, so [&hellip;]","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","_links_to":"","_links_to_target":""},"categories":[8],"tags":[],"class_list":["post-1396","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/posts\/1396","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/comments?post=1396"}],"version-history":[{"count":10,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/posts\/1396\/revisions"}],"predecessor-version":[{"id":1408,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/posts\/1396\/revisions\/1408"}],"wp:attachment":[{"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/media?parent=1396"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/categories?post=1396"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/michaelhoke.name\/oldblog\/wp-json\/wp\/v2\/tags?post=1396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}