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同餘 Congruences

數論. 同餘 Congruences. Definition If an integer m , not zero, devides the difference a-b , we say that a is congruent to b modulo m and write . . 三式等價。 則 。 則 。 則 。 則 。 則 對任意 都成立。. 若 f 是一個整係數 多項式，且 ，則 Ex : , , , . 若且唯若 . , , 則 . for , 若且唯若 . .

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同餘 Congruences

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  1. 數論

  2. 同餘 Congruences • Definition If an integer m, not zero, devides the difference a-b, we say that a is congruent to bmodulo m and write . • .

  3. 三式等價。 • 則。 • 則。 • 則。 • 則。 • 則 對任意都成立。

  4. 若f是一個整係數多項式，且，則 • Ex : , , ,

  5. 若且唯若 . • , , 則 . • for , 若且唯若 .

  6. Definition If , then y is called a residue of x modulo m. • A set is called a complete residue system modulo m if for every integer y there is one and only one such that • 1,2,3,4,5,6,7 is a complete residue system modulo 8.

  7. Euler’s -function is the number of positive integers less than or equal to m that are relatively prime to m. • A reduced residue system modulo m • 1,5 is a reduced residue system modulo 6.

  8. Let . Let be a complete (reduced) residue system modulo m. Then is a complete (redeced) residue system modulo m. • 1,2,3,4,5 is a complete residue system modulo 6. • 5,10,15,20,25 is also a complete residue system modulo 6.

  9. 費馬小定理 Fermat’s theorem • Let p demote a prime. If then . For every integer a, . • For example m=7, , for x is 1,2,3,4,5,6. • .

  10. Euler’s generalization of Fermat’s theorem • If , then .

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