Theoretical Analysis of number theory in algorithm Competitions

I. Basic Concepts

Division algorithm, Division theorem): A ε z, d ε z *, which has a unique integer Q and R, and 0 ≤ r <D0, meet a = D q + R. Q is called the quotient and R is called the remainder. In general, an integer is divided by a positive integer to obtain the quotient and residue ).

Divide exactly: Division B of A. When integer a is divided by a non-zero integer B, the quotient is an integer and the remainder is zero, let's say that a can be divisible by B (or B can be divisible by a), and it is recorded as B |. B is called the divisor of ).

Great common divisor: the largest common divisor of two non-zero integers, A and B, is called D = gcd (A, B ).

Scaled graph equation: a collective name of an Integer Polynomial Equation.

Ii. GCD Euclidean Algorithm

GCD recursion theorem: For any non-negative integer a and any positive integer B, gcd (a, B) = gcd (B, A mod B ). For proof, see the appendix.

Iii. Genus Theorem

Gini theorem: For any integer a, B, linear boosting graph Equation AX + by = m for the X and Y variables ), an integer is obtained only when M is an integer multiple of D = gcd (A, B. For proof, see the appendix.

 

 

Appendix,

Proof of GCD recursion Theorem

Gcd (a, B) = gcd (B, A mod B), First Certificate gcd (a, B) | gcd (B, A mod B ), set d = gcd (a, B), d | A and D | B. Ling

R = (a mod B), then

R = A-QB, Q is a/B Rounded down.

(A mod B) is a linear combination of A and B, so d | (a mod B ).

D | B, d | (a mod B), so d | gcd (B, A mod B ).

In turn, it indicates that gcd (B, A mod B) | gcd (a, B) has the same method and will not be repeated,

Gcd (a, B) = gcd (B, A mod B ).

Proof of the genus Theorem

In AX + by = m,

1. AB = 0, without losing its universality.

B = 0, then

Ax = m. By

D = gcd (A, 0) = A, get

DX = m,

It can be interpreted only when M is an integer multiple of D.

2. AB = 0,

Let A be a linear combination of A and B, that is, a = {XA + Yb | x, y, z },

Make D0 the smallest positive element in a, that is, D0 = min {x | x ε a z *} = x0 A + y0 B,

Next we will first prove that D0 = gcd (a, B), and then construct a group of solutions of the equations to prove the proposition

Adequacy:

Set P to any positive number in A, P = x1a + y1b,

We can prove D0 | P:

Consider P's Division of D0 with remainder,

P = d0q + R, 0 ≤ r <D0, then

R = p-d0q

= X1a + y1b-(x0a + y0b) q

= (X1-qx0) A + (y1-qy0) B,

Because D0 is the smallest positive element in A, R can only be equal to 0, that is, D0 | P.

D0 is the approximate number of positive integers in.

Assume that d Is A and D is the maximum common divisor of any positive integer in a, d | D0, and D0 | D, so D0 = D. In particular

D0 = gcd (| A |, | B |)

= Gcd (A, B ).

Set M = m0d0, D0 = gcd (a, B) = x0 A + y0 B, then

M0 (x0 A + y0 B) = m,

We construct a set of feasible solutions x = m0x0, y = m0y0 for the equations of the sequence.

In fact, the solution can have infinite groups: x = (m0x0 + KB/D0), Y = (m0y0-ka/D0), K, Z.

Necessity: If XA + Yb = m is true, then M is a, D0 | M |, D0 | M.