GF(3125) 本原多项式的数量

hbghlyj

本原多项式的数量公式是什么？
例如number of primitive polynomials of GF(3125) WolframAlpha 结果是 280

hbghlyj

https://www.partow.net/programming/polynomials/index.html 以下是 GF($2^m$) 本原不可约多项式的列表。该列表包含 2 至 32 次的多项式。

Degree 2

x^2 + x^1 + 1
                       

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Degree 3

x^3 + x^1 + 1
                       

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Degree 4

x^4 + x^1 + 1
                       

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Degree 5

x^5 + x^2 + 1
x^5 + x^4 + x^2 + x^1 + 1
x^5 + x^4 + x^3 + x^2 + 1
                       

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Degree 6

x^6 + x^1 + 1
x^6 + x^5 + x^2 + x^1 + 1
x^6 + x^5 + x^3 + x^2 + 1
                       

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Degree 7

x^7 + x^1 + 1
x^7 + x^3 + 1
x^7 + x^3 + x^2 + x^1 + 1
x^7 + x^4 + x^3 + x^2 + 1
x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^7 + x^6 + x^3 + x^1 + 1
x^7 + x^6 + x^4 + x^2 + 1
x^7 + x^6 + x^5 + x^2 + 1
x^7 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
                       

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Degree 8

x^8 + x^4 + x^3 + x^2 + 1
x^8 + x^5 + x^3 + x^1 + 1
x^8 + x^6 + x^4 + x^3 + x^2 + x^1 + 1
x^8 + x^6 + x^5 + x^1 + 1
x^8 + x^6 + x^5 + x^2 + 1
x^8 + x^6 + x^5 + x^3 + 1
x^8 + x^7 + x^6 + x^1 + 1
x^8 + x^7 + x^6 + x^5 + x^2 + x^1 + 1
                       

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Degree 9

x^9 + x^4 + 1
x^9 + x^5 + x^3 + x^2 + 1
x^9 + x^6 + x^4 + x^3 + 1
x^9 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^9 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^9 + x^8 + x^4 + x^1 + 1
x^9 + x^8 + x^5 + x^4 + 1
x^9 + x^8 + x^6 + x^5 + 1
x^9 + x^8 + x^6 + x^5 + x^3 + x^1 + 1
x^9 + x^8 + x^7 + x^2 + 1
x^9 + x^8 + x^7 + x^3 + x^2 + x^1 + 1
x^9 + x^8 + x^7 + x^6 + x^5 + x^1 + 1
x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + 1
                       

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Degree 10

x^10 + x^3 + 1
x^10 + x^4 + x^3 + x^1 + 1
x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^10 + x^8 + x^3 + x^2 + 1
x^10 + x^8 + x^4 + x^3 + 1
x^10 + x^8 + x^5 + x^1 + 1
x^10 + x^8 + x^5 + x^4 + 1
x^10 + x^8 + x^7 + x^6 + x^5 + x^2 + 1
x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^1 + 1
x^10 + x^9 + x^4 + x^1 + 1
x^10 + x^9 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + 1
x^10 + x^9 + x^8 + x^6 + x^5 + x^1 + 1
x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1
                       

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Degree 11

x^11 + x^2 + 1
x^11 + x^5 + x^3 + x^1 + 1
x^11 + x^5 + x^3 + x^2 + 1
x^11 + x^6 + x^5 + x^1 + 1
x^11 + x^7 + x^3 + x^2 + 1
x^11 + x^8 + x^5 + x^2 + 1
x^11 + x^8 + x^6 + x^5 + x^4 + x^1 + 1
x^11 + x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^11 + x^9 + x^4 + x^1 + 1
x^11 + x^9 + x^8 + x^7 + x^4 + x^1 + 1
x^11 + x^10 + x^3 + x^2 + 1
x^11 + x^10 + x^7 + x^4 + x^3 + x^1 + 1
x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1
x^11 + x^10 + x^9 + x^8 + x^3 + x^1 + 1
                       

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Degree 12

x^12 + x^6 + x^4 + x^1 + 1
x^12 + x^9 + x^3 + x^2 + 1
x^12 + x^9 + x^8 + x^3 + x^2 + x^1 + 1
x^12 + x^10 + x^9 + x^8 + x^6 + x^2 + 1
x^12 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1
x^12 + x^11 + x^6 + x^4 + x^2 + x^1 + 1
x^12 + x^11 + x^9 + x^5 + x^3 + x^1 + 1
x^12 + x^11 + x^9 + x^7 + x^6 + x^4 + 1
x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + 1
x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + 1
x^12 + x^11 + x^9 + x^8 + x^7 + x^5 + x^2 + x^1 + 1
x^12 + x^11 + x^10 + x^5 + x^2 + x^1 + 1
x^12 + x^11 + x^10 + x^8 + x^6 + x^4 + x^3 + x^1 + 1
x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1
                       

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Degree 13

x^13 + x^4 + x^3 + x^1 + 1
x^13 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^13 + x^9 + x^8 + x^7 + x^5 + x^1 + 1
x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + 1
x^13 + x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + x^1 + 1
x^13 + x^11 + x^8 + x^7 + x^4 + x^1 + 1
x^13 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^13 + x^12 + x^6 + x^5 + x^4 + x^3 + 1
x^13 + x^12 + x^8 + x^7 + x^6 + x^5 + 1
x^13 + x^12 + x^9 + x^8 + x^4 + x^2 + 1
x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + 1
x^13 + x^12 + x^11 + x^5 + x^2 + x^1 + 1
x^13 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^1 + 1
x^13 + x^12 + x^11 + x^9 + x^5 + x^3 + 1
                       

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Degree 14

x^14 + x^8 + x^6 + x^1 + 1
x^14 + x^10 + x^6 + x^1 + 1
x^14 + x^10 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^14 + x^11 + x^6 + x^1 + 1
x^14 + x^11 + x^9 + x^6 + x^5 + x^2 + 1
x^14 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + 1
x^14 + x^12 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^1 + 1
x^14 + x^12 + x^11 + x^10 + x^9 + x^7 + x^4 + x^3 + 1
x^14 + x^13 + x^6 + x^5 + x^3 + x^1 + 1
x^14 + x^13 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^14 + x^13 + x^11 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^14 + x^13 + x^11 + x^8 + x^5 + x^3 + x^2 + x^1 + 1
x^14 + x^13 + x^12 + x^11 + x^10 + x^7 + x^6 + x^1 + 1
x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^6 + x^5 + 1
                       

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Degree 15

x^15 + x^1 + 1
x^15 + x^4 + 1
x^15 + x^7 + 1
x^15 + x^7 + x^6 + x^3 + x^2 + x^1 + 1
x^15 + x^10 + x^5 + x^1 + 1
x^15 + x^10 + x^5 + x^4 + 1
x^15 + x^10 + x^5 + x^4 + x^2 + x^1 + 1
x^15 + x^10 + x^9 + x^7 + x^5 + x^3 + 1
x^15 + x^10 + x^9 + x^8 + x^5 + x^3 + 1
x^15 + x^11 + x^7 + x^6 + x^2 + x^1 + 1
x^15 + x^12 + x^3 + x^1 + 1
x^15 + x^12 + x^5 + x^4 + x^3 + x^2 + 1
x^15 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^2 + 1
x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
                       

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Degree 16

x^16 + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^16 + x^12 + x^3 + x^1 + 1
x^16 + x^12 + x^7 + x^2 + 1
x^16 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^1 + 1
x^16 + x^13 + x^12 + x^11 + x^7 + x^6 + x^3 + x^1 + 1
x^16 + x^13 + x^12 + x^11 + x^10 + x^6 + x^2 + x^1 + 1
x^16 + x^14 + x^10 + x^8 + x^3 + x^1 + 1
x^16 + x^14 + x^13 + x^12 + x^6 + x^5 + x^3 + x^2 + 1
x^16 + x^14 + x^13 + x^12 + x^10 + x^7 + 1
x^16 + x^15 + x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^9 + x^6 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^9 + x^8 + x^6 + x^4 + x^2 + x^1 + 1
                       

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Degree 17

x^17 + x^3 + 1
x^17 + x^3 + x^2 + x^1 + 1
x^17 + x^5 + 1
x^17 + x^6 + 1
x^17 + x^8 + x^4 + x^3 + 1
x^17 + x^8 + x^7 + x^6 + x^4 + x^3 + 1
x^17 + x^10 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + 1
x^17 + x^12 + x^6 + x^3 + x^2 + x^1 + 1
x^17 + x^12 + x^9 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^17 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^17 + x^14 + x^11 + x^7 + x^5 + x^3 + x^2 + x^1 + 1
x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^5 + x^3 + 1
x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^6 + x^4 + x^2 + x^1 + 1
x^17 + x^16 + x^3 + x^1 + 1
                       

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Degree 18

x^18 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^18 + x^7 + 1
x^18 + x^7 + x^5 + x^2 + x^1 + 1
x^18 + x^8 + x^2 + x^1 + 1
x^18 + x^9 + x^7 + x^6 + x^5 + x^4 + 1
x^18 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^18 + x^9 + x^8 + x^7 + x^6 + x^4 + x^2 + x^1 + 1
x^18 + x^10 + x^7 + x^5 + 1
x^18 + x^10 + x^8 + x^5 + 1
x^18 + x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^18 + x^10 + x^9 + x^3 + 1
x^18 + x^13 + x^6 + x^4 + 1
x^18 + x^15 + x^5 + x^2 + 1
x^18 + x^15 + x^9 + x^2 + 1
                       

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Degree 19

x^19 + x^5 + x^2 + x^1 + 1
x^19 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^6 + x^2 + x^1 + 1
x^19 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
x^19 + x^7 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^8 + x^7 + x^5 + 1
x^19 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^9 + x^8 + x^5 + 1
x^19 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^9 + x^8 + x^7 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
x^19 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^16 + x^13 + x^10 + x^7 + x^4 + x^1 + 1
                       

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Degree 20

x^20 + x^3 + 1
x^20 + x^9 + x^5 + x^3 + 1
x^20 + x^11 + x^8 + x^6 + x^3 + x^2 + 1
x^20 + x^14 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + 1
x^20 + x^17 + x^14 + x^10 + x^7 + x^4 + x^3 + x^2 + 1
x^20 + x^19 + x^4 + x^3 + 1
                       

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Degree 21

x^21 + x^2 + 1
x^21 + x^8 + x^7 + x^4 + x^3 + x^2 + 1
x^21 + x^10 + x^6 + x^4 + x^3 + x^2 + 1
x^21 + x^13 + x^5 + x^2 + 1
x^21 + x^14 + x^7 + x^2 + 1
x^21 + x^14 + x^7 + x^6 + x^3 + x^2 + 1
x^21 + x^14 + x^12 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^21 + x^15 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1
x^21 + x^20 + x^19 + x^18 + x^5 + x^4 + x^3 + x^2 + 1
                       

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Degree 22

x^22 + x^1 + 1
x^22 + x^9 + x^5 + x^1 + 1
x^22 + x^14 + x^13 + x^12 + x^7 + x^3 + x^2 + x^1 + 1
x^22 + x^17 + x^9 + x^7 + x^2 + x^1 + 1
x^22 + x^17 + x^13 + x^12 + x^8 + x^7 + x^2 + x^1 + 1
x^22 + x^20 + x^18 + x^16 + x^6 + x^4 + x^2 + x^1 + 1
                       

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Degree 23

x^23 + x^5 + 1
x^23 + x^5 + x^4 + x^1 + 1
x^23 + x^11 + x^10 + x^7 + x^6 + x^5 + 1
x^23 + x^12 + x^5 + x^4 + 1
x^23 + x^15 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + 1
x^23 + x^16 + x^13 + x^6 + x^5 + x^3 + 1
x^23 + x^17 + x^11 + x^5 + 1
x^23 + x^17 + x^11 + x^9 + x^8 + x^5 + x^4 + x^1 + 1
x^23 + x^18 + x^16 + x^13 + x^11 + x^8 + x^5 + x^2 + 1
x^23 + x^21 + x^7 + x^5 + 1
                       

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Degree 24

x^24 + x^7 + x^2 + x^1 + 1
x^24 + x^21 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^10 + x^9 + x^5 + x^4 + x^1 + 1
x^24 + x^22 + x^20 + x^18 + x^16 + x^14 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + 1
                       

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Degree 25

x^25 + x^3 + 1
x^25 + x^3 + x^2 + x^1 + 1
x^25 + x^11 + x^9 + x^8 + x^6 + x^4 + x^3 + x^2 + 1
x^25 + x^12 + x^4 + x^3 + 1
x^25 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^3 + 1
x^25 + x^17 + x^10 + x^3 + x^2 + x^1 + 1
x^25 + x^18 + x^12 + x^11 + x^6 + x^5 + x^4 + x^3 + 1
x^25 + x^20 + x^5 + x^3 + 1
x^25 + x^20 + x^16 + x^11 + x^5 + x^3 + x^2 + x^1 + 1
x^25 + x^23 + x^21 + x^19 + x^9 + x^7 + x^5 + x^3 + 1
                       

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Degree 26

x^26 + x^6 + x^2 + x^1 + 1
x^26 + x^19 + x^16 + x^15 + x^14 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + 1
x^26 + x^21 + x^18 + x^16 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^3 + 1
x^26 + x^22 + x^20 + x^19 + x^16 + x^13 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^26 + x^22 + x^21 + x^16 + x^12 + x^11 + x^10 + x^8 + x^5 + x^4 + x^3 + x^1 + 1
x^26 + x^23 + x^22 + x^21 + x^19 + x^18 + x^15 + x^14 + x^13 + x^11 + x^10 + x^9 + x^8 + x^6 + x^5 + x^2 + 1
x^26 + x^24 + x^21 + x^17 + x^16 + x^14 + x^13 + x^11 + x^7 + x^6 + x^4 + x^1 + 1
                       

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Degree 27

x^27 + x^5 + x^2 + x^1 + 1
x^27 + x^18 + x^11 + x^10 + x^9 + x^5 + x^4 + x^3 + 1
x^27 + x^22 + x^13 + x^11 + x^6 + x^5 + x^4 + x^3 + 1
x^27 + x^22 + x^17 + x^15 + x^14 + x^13 + x^6 + x^1 + 1
x^27 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^7 + x^5 + 1
x^27 + x^24 + x^19 + x^16 + x^12 + x^8 + x^7 + x^3 + x^2 + x^1 + 1
x^27 + x^24 + x^21 + x^19 + x^16 + x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1
x^27 + x^25 + x^23 + x^21 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^27 + x^25 + x^23 + x^21 + x^20 + x^19 + x^18 + x^16 + x^14 + x^10 + x^8 + x^7 + x^4 + x^3 + 1
                       

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Degree 28

x^28 + x^3 + 1
x^28 + x^13 + x^11 + x^9 + x^5 + x^3 + 1
x^28 + x^18 + x^17 + x^16 + x^9 + x^5 + x^4 + x^3 + 1
x^28 + x^19 + x^17 + x^15 + x^10 + x^6 + x^3 + x^2 + 1
x^28 + x^22 + x^11 + x^10 + x^4 + x^3 + 1
x^28 + x^24 + x^20 + x^16 + x^12 + x^8 + x^4 + x^3 + 1
                       

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Degree 29

x^29 + x^2 + 1
x^29 + x^12 + x^7 + x^2 + 1
x^29 + x^18 + x^14 + x^6 + x^3 + x^2 + 1
x^29 + x^19 + x^16 + x^6 + x^3 + x^2 + 1
x^29 + x^20 + x^11 + x^2 + 1
x^29 + x^20 + x^16 + x^11 + x^8 + x^4 + x^3 + x^2 + 1
x^29 + x^21 + x^5 + x^2 + 1
x^29 + x^23 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1
x^29 + x^24 + x^14 + x^13 + x^8 + x^4 + x^3 + x^2 + 1
x^29 + x^26 + x^5 + x^2 + 1
                       

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Degree 30

x^30 + x^23 + x^2 + x^1 + 1
x^30 + x^24 + x^20 + x^16 + x^14 + x^13 + x^11 + x^7 + x^2 + x^1 + 1
x^30 + x^24 + x^21 + x^20 + x^18 + x^15 + x^13 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^30 + x^25 + x^24 + x^23 + x^19 + x^18 + x^16 + x^14 + x^11 + x^8 + x^6 + x^4 + x^3 + x^1 + 1
x^30 + x^27 + x^25 + x^24 + x^23 + x^22 + x^19 + x^16 + x^12 + x^10 + x^8 + x^7 + x^6 + x^1 + 1
                       

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Degree 31

x^31 + x^3 + 1
x^31 + x^3 + x^2 + x^1 + 1
x^31 + x^13 + x^8 + x^3 + 1
x^31 + x^16 + x^8 + x^4 + x^3 + x^2 + 1
x^31 + x^20 + x^15 + x^5 + x^4 + x^3 + 1
x^31 + x^20 + x^18 + x^7 + x^5 + x^3 + 1
x^31 + x^21 + x^12 + x^3 + x^2 + x^1 + 1
x^31 + x^23 + x^22 + x^15 + x^14 + x^7 + x^4 + x^3 + 1
x^31 + x^25 + x^19 + x^14 + x^7 + x^3 + x^2 + x^1 + 1
x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^7 + x^3 + 1
x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^10 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
                       

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Degree 32

x^32 + x^22 + x^2 + x^1 + 1
x^32 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^1 + 1
x^32 + x^23 + x^17 + x^16 + x^14 + x^10 + x^8 + x^7 + x^6 + x^5 + x^3 + 1
x^32 + x^26 + x^23 + x^22 + x^16 + x^12 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^32 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^17 + x^13 + x^11 + x^10 + x^9 + x^8 + x^7 + x^2 + x^1 + 1
x^32 + x^28 + x^19 + x^18 + x^16 + x^14 + x^11 + x^10 + x^9 + x^6 + x^5 + x^1 + 1
                       

hbghlyj

hbghlyj 发表于 2024-10-16 13:56
例如number of primitive polynomials of GF(3125) WolframAlpha 结果是 280 ...

这个公式是什么？