2的幂

2的幂是指符合${\displaystyle 2^{n}}$型式，而${\displaystyle n}$也是整數的數，也就是底數為2，指數為整數 n的幂。

從1到1024（2⁰ 至 2¹⁰）

在有些情形下，會將${\displaystyle n}$限制在正整數及零的範圍內^[1]，因此2的幂包括1、2以及2自乘多次的乘積^[2]。

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

• ${\displaystyle 2^{n}}$ 
• ${\displaystyle 2\uparrow n}$ 
• ${\displaystyle 2[3]n}$ 
• 2 ^ n
• 2 ** n
• power(2, n)
• 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（${\displaystyle 3=4-1=2^{2}-1}$ ）。
• 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（${\displaystyle 3=2+1=2^{1}+1}$ ）。
• 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的幂的列表（部分）

${\displaystyle 2^{n},n\in [0,101],n\in Z}$ 

2^0

=

1

2^12

=

4,096

2^24

=

16,777,216

2^36

=

68,719,476,736

2^48

=

281,474,976,710,656

2^60

=

1,152,921,504,606,846,976

2^72

=

4,722,366,482,869,645,213,696

2^84

=

19,342,813,113,834,066,795,298,816

2^96

=

79,228,162,514,264,337,593,540,590,336

2^1

=

2

2^13

=

8,192

2^25

=

33,554,432

2^37

=

137,438,953,472

2^49

=

562,949,953,421,312

2^61

=

2,305,843,009,213,693,952

2^73

=

9,444,732,965,739,290,427,392

2^85

=

38.685,626,227,668,133,590,597,632

2^97

=

158,456,325,028,528,675,187,087,900,672

2^2

=

4

2^14

=

16,384

2^26

=

67,108,864

2^38

=

274,877,906,944

2^50

=

1,125,899,906,842,624

2^62

=

4,611,686,018,427,387,904

2^74

=

18,889,465,931,478,580,854,784

2^86

=

77,371,252,455,336,267,181,195,264

2^98

=

316,912,650,057,057,350,374,175,081,344

2^3

=

8

2^15

=

32,768

2^27

=

134,217,728

2^39

=

549,755,813,888

2^51

=

2,251,799,813,685,248

2^63

9,223,372,036,854,775,808

2^75

=

37,778,931,862,957,161,709,568

2^87

=

154,742,504,910,672,534,362,390,528

2^99

=

633,825,300,114,114,700,748,351,602,688

2^4

=

16

2^16

=

65,536

2^28

=

268,435,456

2^40

=

1,099,511,627,776

2^52

=

4,503,599,627,370,496

2^64

=

18,446,744,073,709,551,616

2^76

=

75,557,863,725,914,323,419,136

2^88

=

309,485,009,821,345,068,724,781,056

2^100

=

1,267,650,600,228,229,401,496,703,205,376

2^5

=

32

2^17

=

131,072

2^29

=

536,870,912

2^41

=

2,199,023,255,552

2^53

=

9,007,199,254,740,992

2^65

=

36,893,488,147,419,103,232

2^77

=

151,115,727,451,828,646,838,272

2^89

=

618,970,019,642,690,137,449,562,112

2^101

=

2,535,351,200,456,458,802,993,306,410,752

2^6

=

64

2^18

=

262,144

2^30

=

1,073,741,824

2^42

=

4,398,046,511,104

2^54

=

18,014,398,509,481,984

2^66

=

73,786,976,294,838,206,464

2^78

=

302,231,454,903,657,293,676,544

2^90

=

1,237,940,039,285,380,274,899,124,224

2^7

=

128

2^19

=

524,288

2^31

=

2,147,483,648

2^43

=

8,796,093,022,208

2^55

=

36,028,797,018,963,968

2^67

=

147,573,952,589,676,412,928

2^79

=

604,462,909,817,314,587,353,088

2^91

=

2,475,880,078,570,760,549,798,248,448

2^8

=

256

2^20

=

1,048,576

2^32

=

4,294,967,296

2^44

=

17,592,186,044,416

2^56

=

72,057,594,037,927,936

2^68

=

295,147,905,179,352,825,856

2^80

=

1,208,925,819,614,629,174,706,176

2^92

=

4,951,760,157,141,521,099,596,496,896

2^9

=

512

2^21

=

2,097,152

2^33

=

8,589,934,592

2^45

=

35,184,372,088,832

2^57

=

144,115,188,075,855,872

2^69

=

590,295,810,358,705,651,712

2^81

=

2,417,851,639,229,258,349,412,352

2^93

=

9,903,520,314,283,042,199192,993,792

2^10

=

1,024

2^22

=

4,194,304

2^34

=

17,179,869,184

2^46

=

70,368,744,177,664

2^58

=

288,230,376,151,711,744

2^70

=

1,180,591,620,717,411,303,424

2^82

=

4,835,703,278,458,516,698,824,704

2^94

19,807,040,628,566,084,398,385,987,584

2^11

=

2,048

2^23

=

8,388,608

2^35

=

34,359,738,368

2^47

=

140,737,488,355,328

2^59

=

576,460,752,303,423,488

2^71

=

2,361,183,241,434,822,606,848

2^83

=

9,671,406,556,917,033,397,649,408

2^95

39,614,081,257,132,168,796,771,975,168

2¹ = 2

2² = 4

2⁴ = 16

2⁸ = 256

2¹⁶ = 65,536

2³² = 4,294,967,296

2⁶⁴ = 18,446,744,073,709,551,616

2¹²⁸ = 340,282,366,920,938,463,463,374,607,431,768,211,456

2²⁵⁶ = 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

1. Lipschutz, Seymour. Schaum's Outline of Theory and Problems of Essential Computer Mathematics. New York: McGraw-Hill. 1982: 3. ISBN 0-07-037990-4. 
2. Sewell, Michael J. Mathematics Masterclasses. Oxford: Oxford University Press. 1997: 78. ISBN 0-19-851494-8. 

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