Number Theory remainder Learning

Theorem 3 of the remainder system:

If a, B, c is any three integers, M is a positive integer, and (M, c) = 1, when AC ≡ BC (mod m, there is a limit B (mod m );

Proof: AC cost BC (mod m) <=> (Ac-BC) limit 0 (mod m) <=> (a-B) * C limit 0 (mod m ), if (C, M) = 0, then there must be M | (a-B), that is (a-B) limit 0 (mod m)

<=> A branch B (mod m );

Image Understanding: If M is regarded as a ring runway, and a horse takes a C-meter step at each start point, then steps A and B finally stop at the same position, then there must be M | (a-B ). Because (C, M) = 1 ~ In the M step, the horse must have parked at different m positions on the runway, and also traversed M positions on the runway, therefore, a and B must be an integer multiple of M.

If we regard A and B as different horses and take step C at the same time, it is hard to understand.

Theorem 7 of the remainder system:

Set M to an integer, m> 1, B to an integer and (m, B) = 1. If A1, A2, A3 ,......, Am is a completely residual system of the Modulo M, BA [1], BA [2], BA [3], BA [4],…, Ba [m] also forms a completely residual system of M.

Image Understanding: Because sequence a is a completely residual system, it is in the Mk + C format, because M | MK * C, therefore, the rest is still because of the traversal of the residual series produced by the mutual quality.

Same Theorem 6:

If A, B, C, and D are four integers, and a then B (mod m), C then D (mod m), then AC then BD (mod m );

Understanding: A can be divided into mk1 + H forms, C can also be divided into Mk2 + H forms, and then (mk1 + H) (mk3 + l) Merge (Mk2 + H) (mk4 + l) (mod m)

Proof of ferma's theorem:

If P is a prime number, there must be a number of the remainder (1, 2, 3, 4, 5 ,......, P-1 the mutual quality, we need to traverse all vertices in step 1, that is, return to the original position in P-1, so a ^ (p-1) 1_1 (mod P ).

Principal determination:

If any of the following B statements that satisfy 1 <B <P are true:

B ^ (p-1) 1_1 (mod P)

P must be a prime number.

Proof: If P is a sum, there must be a B that satisfies (B, P )! = 1, then at this time B ^ (p-1 )! Limit 1 (mod P), because B can only traverse to the minimum level of LCM (B, P)/B.

So P must be a prime number.

Number Theory theorem based on information security mathematics:

Chapter 2 -- Tongyu

Complete Residual Series:

Theorem 3: If M is a positive integer, A is an integer that satisfies (a, m) = 1, and B is an arbitrary integer, if X traverses a completely residual system of M, then, ax + B Also traverses a completely residual system of M.

Proof: because (a, m) = 1, we know that a * m is required to return to the origin. That is to say, ax can traverse a completely residual system of M, then + B can traverse the entire residual system, but the position is not correct.

Theorem 4: Let M1 and M2 be the Positive Integers of two reciprocal elements. If X1 and X2 traverse the completely residual systems of M1 and M2 respectively, then m2 * X1 + m1 * X2 traverses the remainder of M1 * M2.

Proof: Because X1 traverses the remainder of M1, and X1 multiplied by an M2 in X1 * m2, it can be assumed that X1 is the smallest non-negative completely remainder, and X2 is the same.

There are M1 * m2 combination methods for X1 and X2, which must be proved to be different from each other. Because M1 and M2 are mutually unique, the origin is returned when X1 is M1 and X2 is m2.

Understanding: a horse that can span 5 or 8 meters in a step can go anywhere on a 5*8 = 40 m runway.

Theorem 5: If p and q are two different prime numbers, and N is their product, there is a unique integer x for any integer c, Y (0 ≤ x <p, 0 ≤ y <q) satisfies QX
+ Py sort C (mod N), because there is a total of p * q options, different options constitute a matrix of p * q.

Simplify the remaining systems:

Theorem 2: Let M be a positive integer. If R1 ,......, Rk is an integer between K = Eular (M) and M, and there are two different integers, then R1 ,...... If rk is distinct from each other, a simplified residual system is formed.

Theorem 3: Let M be a positive integer, and a be an integer that satisfies (a, m) = 1. If X traverses a simplified remainder of M, then AX also traverses a simplified residual system of M.

Because (a, m) = 1, the result of ax traversal is different from each other, because both A and X do not have the common factor of M, so ax also traverses a simplified residual system of M. "Multiplication, surplus ".

Theorem 4: Let M be a positive integer, and a be an integer that satisfies (a, m) = 1, then there is an integer A', 1 ≤ a' ≤ m-1, make a * a' limit 1 (mod m ).

Proof: 1 must belong to a remainder class of M, so ax (1 <x <m) must traverse the point 1 when traversing the remainder of M.

Constructor: Generalized Euclidean algorithm.

Theorem 5: Let M1 and M2 be two integers of the reciprocal element. If X1 and X2 traverse the simplified Remainder of the modulo M1 and modulo m2 respectively, then m2x1 + m1x2 traverses M 1m2 to simplify the rest of the system.

Proof: