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 .

• 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.

• 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.

• 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.

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. • .

Euler’s generalization of Fermat’s theorem

• If , then .