查看“︁2的幂”︁的源代码

{{NoteTA|G1=Math}}
[[Image:Ten octaves visualization.svg|thumb|從1到1024（2<sup>0</sup> 至 2<sup>10</sup>）]]
'''2的幂'''是指符合<math>2^n</math>型式，而<math>n</math>也是[[整數]]的數，也就是[[底數 (指數)|底數]]為[[2]]，指數為整數{{mvar|n}}的[[幂]]。

在有些情形下，會將<math>n</math>限制在正整數及零的範圍內<ref>{{cite book |title=Schaum's Outline of Theory and Problems of Essential Computer Mathematics |url=https://archive.org/details/schaumsoutlineof0000lips_b9v3 |first=Seymour |last=Lipschutz |year=1982 |isbn=0-07-037990-4 |page=[https://archive.org/details/schaumsoutlineof0000lips_b9v3/page/3 3] |publisher=McGraw-Hill |location=New York}}</ref>，因此2的幂包括1、2以及2自乘多次的乘積<ref>{{cite book |title=Mathematics Masterclasses |first=Michael J. |last=Sewell |year=1997 |isbn=0-19-851494-8 |page=78 |publisher=Oxford University Press |location=Oxford}}</ref>。

因為2是[[二進制]]的底數，因此在常出現二進制的[[電腦科學]]中，2的幂也很常見。若將2的幂用二進制表示，會是100…000、0.00…001或是1的形式，類似用[[十進制]]表示[[10的幂]]的情形。

==表示方法==
* <math>2^n</math>
* <math>2 \uparrow n</math>
* <math>2[3]n</math>
* <code>2 ^ n</code>
* <code>2 ** n</code>
* <code>power(2, n)</code>
* 2的n次幂
* 2的n次方

==與2的冪有關的數字==
*比某一个2的幂小1的[[素数]]，在数学上称为[[梅森素数]]；例如数字3是最小的梅森素数（<math>3=4-1=2^2-1</math>）。
*比某一个2的幂大1的素数，在数学上称为[[费马素数]]；如数字3也是最小的费马素数（<math>3=2+1=2^1+1</math>）。
*一个以2的幂为分母的分数称为[[二进有理数]]。
*可以表示为连续正整数和的数称为[[礼貌数]]，2的幂不會是礼貌数。

== 2的幂列表 ==
<math>2^n, n\in [0,63], n\in Z</math>

{| class="wikitable" style="text-align:center"
|- style="background:#f9f9f9;" 
|'''2<sup>0</sup>'''  || = ||align="right"| '''[[1]]'''
|bgcolor="white" rowspan=16|
|'''2<sup>16</sup>'''|| = ||align="right"| '''[[65536|65,536]]'''
|bgcolor="white" rowspan=16|
|'''2<sup>32</sup>'''|| = ||align="right"| '''4,294,967,296'''
|bgcolor="white" rowspan=16|
|'''2<sup>48</sup>'''|| = ||'''281,474,976,710,656'''
|-----
|2<sup>1</sup>  || = ||align="right"| [[2]]
|2<sup>17</sup>|| = ||align="right"| 131,072
|2<sup>33</sup>|| = ||align="right"| 8,589,934,592
|2<sup>49</sup>
|=
|562,949,953,421,312
|-----
|2<sup>2</sup>  || = ||align="right"| [[4]]
|2<sup>18</sup>|| = ||align="right"| 262,144
|2<sup>34</sup>|| = ||align="right"| 17,179,869,184
|2<sup>50</sup>
|=
|1,125,899,906,842,624
|-----
|2<sup>3</sup>  || = ||align="right"| [[8]]
|2<sup>19</sup>|| = ||align="right"| 524,288
|2<sup>35</sup>|| = ||align="right"| 34,359,738,368
|2<sup>51</sup>
|=
|2,251,799,813,685,248
|-----
|'''2<sup>4</sup>'''  || = ||align="right"| '''[[16]]'''
|'''2<sup>20</sup>'''|| = ||align="right"| '''1,048,576'''
|'''2<sup>36</sup>'''|| = ||align="right"| '''68,719,476,736'''
|'''2<sup>52</sup>'''
|=
|'''4,503,599,627,370,496'''
|-----
|2<sup>5</sup>  || = ||align="right"| [[32]]
|2<sup>21</sup>|| = ||align="right"| 2,097,152
|2<sup>37</sup>|| = ||align="right"| 137,438,953,472
|2<sup>53</sup>
|=
|9,007,199,254,740,992
|-----
|2<sup>6</sup>  || = ||align="right"| [[64]]
|2<sup>22</sup>|| = ||align="right"| 4,194,304
|2<sup>38</sup>|| = ||align="right"| 274,877,906,944
|2<sup>54</sup>
|=
|18,014,398,509,481,984
|-----
|2<sup>7</sup>  || = ||align="right"| [[128]]
|2<sup>23</sup>|| = ||align="right"| 8,388,608
|2<sup>39</sup>|| = ||align="right"| 549,755,813,888
|2<sup>55</sup>
|=
|36,028,797,018,963,968
|-----
|'''2<sup>8</sup>'''  || = ||align="right"| '''[[256]]'''
|'''2<sup>24</sup>'''|| = ||align="right"| '''16,777,216'''
|'''2<sup>40</sup>'''|| = ||align="right"| '''1,099,511,627,776'''
|'''2<sup>56</sup>'''
|=
|'''72,057,594,037,927,936'''
|-----
|2<sup>9</sup>  || = ||align="right"| [[512]]
|2<sup>25</sup>|| = ||align="right"| 33,554,432
|2<sup>41</sup>|| = ||align="right"| 2,199,023,255,552
|2<sup>57</sup>
|=
|144,115,188,075,855,872
|-----
|2<sup>10</sup> || = ||align="right"| [[1024|1,024]]
|2<sup>26</sup>|| = ||align="right"| 67,108,864
|2<sup>42</sup>|| = ||align="right"| 4,398,046,511,104
|2<sup>58</sup>
|=
|288,230,376,151,711,744
|-----
|2<sup>11</sup> || = ||align="right"| [[2000#2048|2,048]]
|2<sup>27</sup>|| = ||align="right"| 134,217,728
|2<sup>43</sup>|| = ||align="right"| 8,796,093,022,208
|2<sup>59</sup>
|=
|576,460,752,303,423,488
|-
|'''2<sup>12</sup>'''
|=
|'''[[4096|4,096]]'''
|'''2<sup>28</sup> '''
|=
|'''268,435,456'''
|'''2<sup>44</sup>'''
|=
|'''17,592,186,044,416'''
|'''2<sup>60</sup>'''
|=
|'''1,152,921,504,606,846,976'''
|-
|2<sup>13</sup>
|=
|[[8192|8,192]]
|2<sup>29</sup>
|=
|536,870,912
|2<sup>45</sup>
|=
|35,184,372,088,832
|2<sup>61</sup>
|=
|2,305,843,009,213,693,952
|-
|2<sup>14</sup>
|=
|[[16384|16,384]]
|2<sup>30</sup>
|=
|1,073,741,824
|2<sup>46</sup>
|=
|70,368,744,177,664
|2<sup>62</sup>
|=
|4,611,686,018,427,387,904
|-
|2<sup>15</sup>
|=
|[[32768|32,768]]
|2<sup>31</sup>
|=
|2,147,483,648
|2<sup>47</sup>
|=
|140,737,488,355,328
|2<sup>63</sup>
|=
|9,223,372,036,854,775,808
|}

== 2的2的幂次方列表 ==
: 2<sup>2<sup>0</sup></sup> = 2<sup>1</sup> = [[2]]
: 2<sup>2<sup>1</sup></sup> =2<sup>2</sup> = [[4]]
: 2<sup>2<sup>2</sup></sup> =2<sup>4</sup> = [[16]]
: 2<sup>2<sup>3</sup></sup> =2<sup>8</sup> = [[256]]
: 2<sup>2<sup>4</sup></sup> =2<sup>16</sup> = [[65536|65,536]]
: 2<sup>2<sup>5</sup></sup> =2<sup>32</sup> = 4,294,967,296
: 2<sup>2<sup>6</sup></sup> =2<sup>64</sup> = 18,446,744,073,709,551,616
: 2<sup>2<sup>7</sup></sup> =2<sup>128</sup> = 340,282,366,920,938,463,463,374,607,431,768,211,456
: 2<sup>2<sup>8</sup></sup> =2<sup>256</sup> = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936
==參考資料==
{{reflist}}
==相關條目==
*[[二進制]]
*[[等比数列]]
*[[以2爲底的對數]]
*[[Octave (電子學)]]
*{{le|無和數列|Sum-free sequence}}
*{{le|勾德數列|Gould's sequence}}
*[[2048 (遊戲)|2048]]（與2的冪有關的電子遊戲）
*[[国际象棋盘与麦粒问题]]
*[[汉诺塔|漢諾塔]]
*[[淘汰制]]
*[[音符]]
{{級數}}

[[Category:数]]
[[Category:整數]]
[[Category:整數數列]]
[[Category:二进制算术]]