信息学跟数学基础.doc

第一章 有关数论的算法 1.1 最大公约数与最小公倍数1.2 有关素数的算法1.3 方程ax+by=c的整数解及应用 1.4 求a^b mod n 1.1最大公约数与最小公倍数 1.算法1: 欧几里德算法求a，b的最大公约数?function gcd(a,b:longint):longint;?begin?if b=0 then gcdd:=a?else gcd:=gcd(b,a mod b);?end;2.算法2:最小公倍数acm=a*b div gcd(a,b); 3.算法3:扩展的欧几里德算法，求出gcd(a,b)和满足gcd(a,b)=ax+by的整数x和y?function exgcd(a,b:longint;var x,y:longint):longint;vart:longint;beginif b=0 then??begin? result:=a;? x:=1;? y:=0;?endelse?begin?result:=exgcd(b,a mod b,x,y);?t:=x;?x:=y;?y:=t-(a div b)*y;?end;end; （理论依据：gcd(a,b)=ax+by=bx1+(a mod b)y1=bx1+(a-(a div b)*b)y1=ay1+b(x1-(a div b)*y1)) 1. 2有关素数的算法 1.算法4:求前n个素数: program BasicMath_Prime;constmaxn=1000;varpnum,n:longint;?p:array[1..maxn] of longint;function IsPrime(x:longint):boolean;var i:integer;beginfor i:=1 to pnum do?if sqr(p[i])=x then? begin?? if x mod p[i]=0 then???? begin????? IsPrime:=false;?????? exit;???? end;?end? ?else? begin?? IsPrime:=true;?? exit;?end;IsPrime:=true;end;procedure main;var x:longint;beginpnum:=0;x:=1;while(pnumn) dobegin?inc(x);?if IsPrime(x) then? begin?? inc(pnum);?? p[pnum]:=x;? end;end; end;procedure out;var i,t:integer;beginfor i:=1 to n do?begin ?write(p[i]:5);t:=t+1; ?if t mod 10=0 then writeln; ?end;end;beginreadln(n); main;out;end. 2.算法5:求不大于n的所有素数 program sushu3;const maxn=10000;vari,k,n:integer;a:array[1..maxn] of integer;beginreadln(n);for i:=1 to n do a[i]:=i;a[1]:=0;i:=2;while in dobegin?k:=2*i;?while k=n do? begin? a[k]:=0;? k:=k+i;? end;?i:=i+1;?while (a[i]=0) and (in) do i:=i+1;end;k:=0;for i:=1 to n do?if a[i]0? then?????? begin?????? write(a[i]:5); k:=k+1;?????? if k mod 10 =0 then writeln;?????? endend. 3.算法6:将整数分解质因数的积 program BasicMath_PolynomialFactors;constmaxp=1000;varpnum,n:longint;num,p:array[1..maxp] of longint;procedure main;var x:longint;begin?fillchar(num,sizeof(num),0);?fillchar(p,sizeof(p),0);?pnum:=0;?x:=1;?while(n1) do? begin? inc(x);? if n mod x=0 then??? beg