# The simplest fractional Farey sequence of number theory and the +poj3374 of the simple fraction

Source: Internet
Author: User

Mr Farley series

definitions and theorems

Definition one: The simplest fraction (also known as either a fraction or an irreducible fraction). If P,q's greatest common divisor is 1, we call the score p/q the simplest fraction.

Definition two: True score, if P,q is a positive integer, 0<p/q<1, we say p/q is the true fraction

Theorem:

Score A/b, C/D is the simplest true score (also 0/1 or 1/1) and A/b <c/d, there are

1) Number (A+C)/(B+D) is a minimal fraction

2) A/b < (A+C)/(B+d) < C/D

Proof slightly

Definition of Mr Farley series:

The method of derivation of the method of thunder sequence:

In the application theorem (2), if A/b and C/D are a sequence of Valeo, then p/q = (a+c)/(B+D) can be inserted between them (provided that the Q value is less than or equal to the given n value) so it can be constructed in two minutes, until not for example

N = 5:

Step1: Prepare two numbers 0/1, 1/1 as the first and last element of the entire Valeo sequence 0/1, 1/1

Step2: Insert 1 numbers in the middle of two numbers 1/2, into 0/1, 1/1

Step3: Insert 1 numbers in the middle of each pair of adjacent pairs, becoming 0/1, 1/3, 2/3, 1/1

step4: Insert 1 numbers in the middle of each pair of adjacent pairs, becoming 1/4,1/3,   , 1/2,    2/3,   3/4 1/1

Between STEP5:0/1 and 1/4 and 3/4 and 1/1 can still be inserted 1 numbers, and the number of inserted denominator is not greater than 5 0/1, 1/5 , Quarter , 1/3, 2/5, , 3/5, 2/3, 3/4,< /c7> 4/5 , 1/1

At this point, the sequence contains all of the least-true fractions with a denominator of not more than 5, and the individual fractions are arranged in ascending order.

`1#include <iostream>2#include <cstdio>3#include <cstring>4 #defineMAXN 80000005 using namespacestd;6 intn,t;7 intct=2;8 intTX[MAXN],TY[MAXN];9 voidFarey (intAintBintCintd)Ten { One     if(B+d > N)return ; A     /*if (b+d = = N) -     { - printf ("%d/%d", a+c,b+d); the if (a+c! = N-1) printf (","); - return; -     }*/ -Farey (a,b,a+c,b+d); +     //printf ("%d/%d,", a+c,b+d); -tx[ct]=a+C; +ty[ct++]=b+D; AFarey (a+c,b+d,c,d); at } - intMain () - { -scanf"%d%d",&n,&t); -tx[1]=0; -ty[1]=1; inFarey (0,1,1,1); -tx[ct]=1; toty[ct]=1; +      for(intI=1; i<=t;i++) -     { the         intm; *scanf"%d",&m); \$         if(m>CT)Panax Notoginsengprintf"No solution\n"); -         Else theprintf"%d/%d\n", Tx[m],ty[m]); +     } A     return 0; the}`

The simplest fractional Farey sequence of number theory and the +poj3374 of the simple fraction

Related Keywords:

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

## A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

• #### Sales Support

1 on 1 presale consultation

• #### After-Sales Support

24/7 Technical Support 6 Free Tickets per Quarter Faster Response

• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.