An array of hooks and its application uva106

Tick array

If the ternary group (A,B,C) satisfies a^2 + b^2 = C^2 's tick array, is there an infinite number of tick arrays?

The answer is yes, multiply the triples by D, you can get the new triples (DA,DB,DC) (DA) ^2 + (db) ^2 = (DC) ^2--(a^2+b^2) * d^2 =c^2 * d^2

The value of D is arbitrary, so there are multiple tick arrays

An array of origin hooks and strands

The source tick array is a ternary group (A,B,C), where a,b,c only exists common factor 1 and satisfies a^2 + b^2 = c^2

Accumulating data: Some of the following source tick arrays

(3,4,5), (5,12,13), (8,15,17), (7,24,25)

(20,21,29), (9,40,41), (12,35,37), (11,60,61)

Analyze data:

It can be inferred from these data that there may be a conclusion that the parity of a, B is different, and that C is always odd

Proof: If A and B are even, then C is definitely an even number, then there is a common factor of 2, so the ternary group is not the origin of the A,b,c

If A and B are odd, then C is necessarily an even number. Set a=2*x+1, B=2*y+1,c=2*z bring it into a^2 + b^2 = c^2

Get (2x+1) ^2 + (2y+1) ^2 = (2z) ^2--4x^2+4x+4y^2+4y+2=4z^2

The left side of the equation is an odd number and the right is an even number, so the above hypothesis is not true

　So a, B is an even number, an odd number, and C is an odd number.

　　

How to quickly get a certain range of source tick array

(A,B,C) is the source hook array, and A is odd, B is even, C is odd, then can be factorization,

a^2 = c^2-b^2 = (c-b) (c+b)

3^2 = (5-4) (5+4) = 1 * 9

15^2 = (17-8) (17+8) = 9 * 25

32^2 = (37-12) (37+12) = 25 * 49

As if C-b and c+b always squared,

Proof: Hypothesis (c-b)%d==0 && (c+b)%d==0, then according to the nature of the mold, there is (c+b) + (c-b) = 2c, 2c%d==0, and (c+b)-(c-b) =2b,2b%d==0,

That is, D is divisible by 2b and 2c, because B and C do not have common factor, so d can only be 1 or 2, by (C+b) (C-b) =a^2 is also a multiple of D, and a is odd, so a^2 is odd, so d can only take value 1

So (C-b) and (C+b), and (C-b) (c+b) =a^2, namely (C-b) and (C+b) and is the sum of squares, This situation only occurs when (C-b) and (C+B) itself is the sum of squares

Set c+b=s^2,c-b=t^2, where s>t>=1 was no common factor of odd

To solve the equation

uva106 http://uva.onlinejudge.org/index.php?option=com_onlinejudge&itemid=8&page=show_problem& category=115&problem=42&mosmsg=submission+received+with+id+15454112

Give us a certain n, want us to ask how many 1->n within the source hook array, and how many do not belong to the hook array of numbers

1#include <stdio.h>2#include <string.h>3#include <stdlib.h>4#include <algorithm>5#include <iostream>6#include <queue>7#include <stack>8#include <vector>9#include <map>Ten#include <Set> One#include <string> A#include <math.h> - using namespacestd; - #pragmaWarning (disable:4996) thetypedefLong LongLL;  - Const intINF =1<< -; - /* -  + */ - BOOLvis[1000000+Ten]; + intans1[1000000+Ten]; A intgcdintAintb) at { -     if(b = =0) -         returnA; -     returnGCD (b, a%b); - } - intMain () in { -     intS, T, A, B, C; to     intCNT =0, N; +      while(SCANF ("%d", &n)! =EOF) -     { theCNT =0; *memset (ANS1,0,sizeof(ans1)); $memset (Vis,0,sizeof(Vis));Panax Notoginseng         intm = sqrt (2*n); -          for(s =3; S <= m; s + =2) the         { +              for(t =1; T < S; T + =2) A             { the                 if(GCD (S, t)! =1)Continue; +A = S *T; -b = (s*s-t*t)/2; $c = (s*s + t*t)/2; $                 if(c>n) Break; -                 if(ll) A*a + (ll) B*b = = (LL) c*c) -                 { the                     //printf ("%d%d%d\n", A, B, c); -ans1[c]++;Wuyi                      for(inti = a, J = b, k = c; K <= N; i + = A, J + = B, k + =c) theVis[i] = vis[j] = Vis[k] =true; -                 } Wu                 Else if(ll) A*a + (LL) B*b < (LL) c*c) -                      Break; About             } $         } -  -          for(inti =1; I <= N; ++i) -         { AAns1[i] + = ans1[i-1]; +             if(!Vis[i]) thecnt++; -         } $printf"%d%d\n", Ans1[n], CNT); the     } the     return 0; the}

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An array of hooks and its application uva106