# reciprocity theorem

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### Bzoj 1951: [Sdoi2010] Ancient pig literature (Lucas theorem &amp;&amp; China remainder theorem &amp;&amp; Fermat theorem)

1951: [Sdoi2010] Ancient pig wen Seeking G^sigma{c (N, i), I | N} The value of mod m, where m = 999911659.M is a prime number, so according to Fermat theorem g^ (M-1) ≡1 (mod M).Then G^sigma{c (N, i), I | n}≡g^ (Sigma{c (N, i), I | n} mod (M-1)) ≡g^sigma{c (n, i) mod (M-1), I | N} (mod M). So we only need C (N, K) mod (M-1) and then accumulate.Note: M-1 = 2 * 3 * 4679 * 35617.So we can getP≡A1 (mod 2)P≡A2 (mod 3)P≡A3 (mod 4679)P≡A4 (mod 35617)After t

### Miller_rabin prime number test [Fermat Theorem] [quadratic probe theorem] [same remainder formula] [Wilson Theorem]

by c = 2 (n = 10) 2-32 (mod 10) Wilson's Theorem When and only when P is a prime number: (p-1 )! Interval-1 (mod P ). In short, the number between {2 .. P-2} can be paired by two. The product is the same as the 1-module p. Therefore, the concatenation part can be changed to 1, with only 1 * (PM) left. The last formula is used to verify the result. [But why can we pair them? I haven't understood it yet. I'll understand it later.] WilsonThe

### "Lucas theorem/Fermat theorem/Chinese remainder theorem/extended Euclid" [Bzoj 1951] Ancient pig text

[Description] Ask[Solution]Easy to get,So, the focus is on how to askIf P-1 is a prime number, we can enumerate all d with the time of sqrt (n) and calculate the sum by the Lucas theorem respectively.But we find that p-1=2*3*4679*35617 is not a prime number, so does Lucas ' theorem work? No, we can work out the model of 2, 3, 4679, 35617, and write four congruence equations, then solve them with grandson's

### The existence theorem of inverse function of mathematical analysis, continuity theorem and derivation theorem

The existence theorem of inverse function If the function y=f (x), x∈df y = f (x), x \in D_f is strictly monotonically increasing (decreasing), then there is its inverse functionX=f−1 (y): rf→x x = f^{-1} (y): R_f \rightarrow X, and f−1 (y) f^{-1} (Y) is also strictly monotonically increased (reduced). Proof: It is advisable to set y=f (x), x∈df y = f (x), x \in d_f strictly monotonic increments, ∀x1,x2∈df,x1∀y1,y2∈df−1

### Euler's theorem and ferma's theorem, discrete logarithm's Theorem

Ferma's TheoremIs a theorem in number theory: Assume thatAIs an integer,PIs a prime number, so it is a multiple of P, which can be expressed IfANoPThis theorem can also be written as a multiple This writing method is more commonly used. Euler's Theorem(Also calledFerma-Euler's TheoremOrEuler's Function Theor

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### "BZOJ1951" "SDOI2010" Ancient pig Lucas theorem, Chinese remainder theorem, exgcd, Fermat theorem

, when k equals a certain value, the number of characters of the pig in the face is n/k.。 However, it is also quite a lot to keep n/k from n characters. Ipig predicts that if all the possible k's cases add up to p, then the cost of studying ancient writings will be the P-th side of G. Now he wants to know what the cost of studying ancient writings in the Pig Kingdom is. Since Ipig thinks this number can be astronomical, you just need to tell him that the answer is divided by the remainder of 999

### "Everyday Learning" multiplication inverse &amp;&amp; Euler theorem &amp;&amp; fermat theorem &amp;&amp; Euler function application &amp;&amp; University Bully

to multiplyTheoretical basis:f/a mod C =?if present a*x = 1 (mod C)//Why is 1, because the above multiplication (?) * (1) (mod 7) should be 1 This number will not changeso the 2 sides multiply at the same time, get F * X =? (mod C)Now we know that dividing by a number modulo equals multiplying the number of the inverse of the modulo is the next step is to find the inverse of the inverse element is actually to seek to make a*x = 1 (mod C) established X, where a is the divisor in the DivisionEule

### Morley theorem: UVa 11178 Morley ' s theorem

Molly's theorem (Morley's theorem), also known as the three-point theorem of the Morey angle. The three inner corners of the triangle are divided into three equal points, and the two three-point lines near one side intersect to get an intersection, so that three intersections can form a positive triangle. This triangle is often called the Molly Triangle.

### Euler's theorem, extended Euler's Theorem and Its Application (Euler's power reduction method)

Abstract: This paper mainly introduces the Euler's Theorem in number theory, and then introduces the extension and application of the Euler's theorem. Combined with examples, it shows how to use the extended Euler's Theorem to implement power reduction modulo. In number theory, Euler's theorem,(Also called ferma- Euler

### The Nyquist theorem and the re-understanding of Shannon's theorem

the Nyquist theorem and the re-understanding of Shannon's theorem @ (computer network) About the code element, first review the concept: http://blog.csdn.net/u011240016/article/details/53333682 Nyquist theorem Under ideal low-pass conditions – no noise, limited bandwidth channels, and a limit code element rate of 2W baud.W is the channel bandwidth, in Hz. The nu

### Examples of Chinese Remainder Theorem algorithm implemented by Python and python theorem Algorithm

Examples of Chinese Remainder Theorem algorithm implemented by Python and python theorem Algorithm This example describes the Chinese Remainder Theorem algorithm implemented by Python. We will share this with you for your reference. The details are as follows: Chinese Remainder Theorem-CRT: Also known as Sun Tzu's

### [Poj1548] robots Dilworth theorem (partial sequence set theorem 2)

Label: Dilworth theorem poj1548 greedy Question: There are some places with garbage, so that the robot can only go to the right or down from the upper left corner, and ask how many times it can clean up all the garbage, Question: First, it is a classic question of network flow, or a bipartite graph-minimum path coverage. But now, after all, we are doing some greed. This question uses a greedy correlation t

### Explanation of extended Chinese Remainder Theorem and explanation of Chinese Theorem

Explanation of extended Chinese Remainder Theorem and explanation of Chinese TheoremPreface Before reading this article, we recommend that you first learn the Chinese Remainder Theorem. In fact, it doesn't matter if you don't learn it. After all, there is no relationship between the two.Extended CRT We know that the Chinese Remainder Theorem is used to solve the

### CSU 1805 three capitals (Matrix tree theorem +best theorem)

http://acm.csu.edu.cn/csuoj/problemset/problem?pid=1805Test instructionsA and B have a side, a and G have a B-bar, and B and G have a C-side. Now go through all the edges from point a, and then go back to point a, and ask how many ways there are.Ideas:16 Hunan Province, the title of the problem is to find the number of Euro-loop, and the count of spanning tree has a certain connection.First of all, the magic of the best theorem, which is what the devi

### poj1006 China remainder theorem &amp;&amp; analysis of Chinese remainder theorem

The idea of POJ 1006 is not very difficult and can be transformed into mathematical formula:Now num is the number of days from the next same day to the beginningP,e,i,d title in the set!Then you can get three formulas: (num + d)% = = p, (num + d)% = = e; (num + d)% = = i;P,e,i,d is our input, then we need to find Num can, for convenience, we will num+d temporarily as a whole! make x = num + D;namely: x% = = p; x% = = e; x% = = i; xWhat to do? This involves the so-called " Chinese remainder

### The only decomposition theorem (arithmetic basic theorem) and its application

arithmetic basic theorem : any one natural number greater than 1 N, if n is not prime , then n can be uniquely decomposed into the product of finite prime numbersN = p1^a1 * P2^A2 * p3^a3 * ... * pn^an(Wherein P1, p2 、... pn is n factor, A1, A2 、... 、 an exponent of the factor respectively)Such decomposition is called the standard decomposition of NApplication:(1) A positive integer greater than 1 N, if its standard decomposition is:N = p1^a1 * P2^A2

### Closed interval set theorem (Nested intervals theorem) explanation 1

①②, the limit and the unequal relationship are used here.③ if a≠b, then there will be no $\lim _{n\rightarrow \infty}\left| I_n \right| =0$④ If there's still a little C inInside, then there will be no $\lim _{n\rightarrow \infty}\left| I_n \right| =0$For a closer look at the closed interval set theorem (Nested intervals theorem), see the explanation 2:http://www.cnblogs.com/imath/p/6260953.htmlClosed interv

### HDU 5446 Unknown Treasure Lucas theorem + Chinese remainder theorem

combination on a line.Sample Input19 5 23 5Sample OUTPUT6Source2015 ACM/ICPC Asia Regional Changchun OnlineRecommendhujie | We have a carefully selected several similar problems for you:5842 5841 5840 5839 5838 Analysis: According to Lucas solution each I:C (n,m)%pi, and then according to the The state surplus theorem integrates these results. will be able to get answers. Note that the Chinese remainder theorem

### Wilson's theorem, infinite monkey's Theorem

Wilson's theorem, infinite monkey's Theorem Wilson's theorem proves that if p is a prime number, p can be divisible by (p-1 )! + 1 First: First test the strange Value If p = 2, true; If p = 3, true; Then we will start to study the situation where p> = 5 Second: let's prove several conclusions first. Set A = {2, 3, 4 ,......, P-2} Note B = {a, 2a, 3a ,......, (P

### Ferma's theorem and ferma's Theorem

Ferma's theorem and ferma's Theorem When p is a prime number, any integer x has xp limit x (mod p). This theorem is called The FeiMa small theorem. Where if x cannot be divisible by p, we have XP-1 limit 1 (mod p ). After Deformation of this formula, we can get the A-1 mod p (mod p), so we can obtain the multiplicati

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