File:Znam-2-3-11-23-31.svg
Summary
Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of squares has k squares of side length 1/k, for some k in the set {2,3,11,23,31,47058}; for instance the first row has two squares of side length 1/2. Thus, each row of squares has area 1/k, and all six rows together exactly cover a unit square. The bottom row, with 47058 squares of side length 1/47058, would be too small to see in the figure, and is not shown. Sets of integers such that <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cea92d7b5d7eb3846a8259d154b175a526115f" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.338ex; width:23.238ex; height:3.843ex;" alt="{\displaystyle 1=\sum 1/x_{i}+\prod 1/x_{i}}">, such as the set {2,3,11,23,31} used to construct this figure, correspond to solutions of <a href="https://en.wikipedia.org/wiki/Zn%C3%A1m%27s_problem" class="extiw" title="en:Znám's problem">Znám's problem</a>. As all numbers in the set {2,3,11,23,31} are prime, their product 47058 is a <a href="https://en.wikipedia.org/wiki/primary_pseudoperfect_number" class="extiw" title="en:primary pseudoperfect number">primary pseudoperfect number</a>.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 21:15, 15 January 2017 | | 396 × 396 (7 KB) | 127.0.0.1 (talk) | <p>Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of squares has k squares of side length 1/k, for some k in the set {2,3,11,23,31,47058}; for instance the first row has two squares of side length 1/2. Thus, each row of squares has area 1/k, and all six rows together exactly cover a unit square. The bottom row, with 47058 squares of side length 1/47058, would be too small to see in the figure, and is not shown. Sets of integers such that <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>1</mn><mo>=</mo><mo>∑<!-- ∑ --></mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msub><mo>+</mo><mo>∏<!-- ∏ --></mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 1=\sum 1/x_{i}+\prod 1/x_{i}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cea92d7b5d7eb3846a8259d154b175a526115f" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -1.338ex; width:23.238ex; height:3.843ex;" alt="{\displaystyle 1=\sum 1/x_{i}+\prod 1/x_{i}}"></span>, such as the set {2,3,11,23,31} used to construct this figure, correspond to solutions of <a href="https://en.wikipedia.org/wiki/Zn%C3%A1m%27s_problem" class="extiw" title="en:Znám's problem">Znám's problem</a>. As all numbers in the set {2,3,11,23,31} are prime, their product 47058 is a <a href="https://en.wikipedia.org/wiki/primary_pseudoperfect_number" class="extiw" title="en:primary pseudoperfect number">primary pseudoperfect number</a>. </p> |
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