Sz\303\241m\303\255t\303\263g\303\251pes sz\303\241melm\303\251let J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak

restart; with(numtheory);

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

[i$i=1..20]; evalf(map(i->log[2](mersenne([i])+1),%));

NzYiIiIiIiMiIiQiIiUiIiYiIiciIigiIikiIioiIzUiIzYiIzciIzgiIzkiIzoiIzsiIzwiIz0iIz4iIz8=

NzYkIiIjIiIhJCIiJEYlJCIiJkYlJCIiKEYlJCIjOEYlJCIjPEYlJCIjPkYlJCIjSkYlJCIjaEYlJCIjKilGJSQiJDIiRiUkIiRGIkYlJCIkQCZGJSQiJDInRiUkIiV6N0YlJCIlLkFGJSQiJSJHI0YlJCIlPEtGJSQiJWBVRiUkIiVCV0Yl

# # This procedure approximate Cs calculating the product # for primes below x. # Cs:=proc(s::posint,x::posint) local P,p; P:=1.; p:=nextprime(s); while p<x do P:=P*(1-s/p)/(1-1/p)^s; p:=nextprime(p) od; P end;

Zio2JCdJInNHNiJJJ3Bvc2ludEclKnByb3RlY3RlZEcnSSJ4R0YmRic2JEkiUEdGJkkicEdGJkYmRiZDJj5GLCQiIiIiIiE+Ri0tSSpuZXh0cHJpbWVHNiRGKEkoX3N5c2xpYkdGJjYjRiU/KEYmRjFGMUYmMkYtRipDJD5GLCooRixGMSwmRjFGMSomRiVGMUYtISIiRkBGMSksJkYxRjEqJEYtRkBGQEYlRkA+Ri0tRjU2I0YtRixGJkYmRiY=

Cs(2,10); Cs(2,100); Cs(2,1000); Cs(2,10000); Cs(2,100000); Cs(2,1000000);

JCIrKVxQZiRvISM1

JCIrWTN4OG0hIzU=

JCIrWnVYLW0hIzU=

JCIrdUhvLG0hIzU=

JCIrR01pLG0hIzU=

JCIrbSQ9O2cnISM1

f1:=h->(3.+30*h)*2.^38880.-1; f2:=f1+2; f2(0); g:=h->1/ln(f1(h))/ln(f2(h));

Zio2I0kiaEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmJCIiJCIiISIiIiwkKiYiI0lGL0YkRi9GL0YvRi8pJCIiI0YuJCImISkpUUYuRi9GL0YvISIiRiVGJUYl

LCZJI2YxRzYiIiIiIiIjRiU=

JCIrR0cocEwkIiYmcDY=

Zio2I0kiaEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkjZjFHRiVGIyEiIi1GKzYjLUkjZjJHRiVGI0YyRiVGJUYl

2.^27/6*(g(0)+4*g(2.^26)+g(2.^27));

JCIrZkVgWD0hIzU=

Cf1f2:=C2*(1-1/3)^2/(1-2/3)*(1-1/5)^2/(1-2/5)/(1-1/2)^2/(1-1/3)^2/(1-1/5)^2;

LCQqJiIjPyIiIkkjQzJHNiJGJUYl

%%*20*0.66016;

JCIrIW8kcE9DISIq

f1:=h->(5775.+30030*h)*2.^171960.-1; f2:=f1+2; f2(0); g:=h->1/ln(f1(h))/ln(f2(h));

Zio2I0kiaEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmJCIldmQiIiEiIiIsJComIiZJKyRGL0YkRi9GL0YvRi8pJCIiI0YuJCInZz48Ri5GL0YvRi8hIiJGJUYlRiU=

LCZJI2YxRzYiIiIiIiIjRiU=

JCIrOEwhKnl2IiZmPCY=

Zio2I0kiaEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkjZjFHRiVGIyEiIi1GKzYjLUkjZjJHRiVGI0YyRiVGJUYl

2.^33/6*(g(0)+4*g(2.^32)+g(2.^33));

JCIrQ3hKVmchIzU=

C2*(1-1/3)^2/(1-2/3)*(1-1/5)^2/(1-2/5)*(1-1/7)^2/(1-2/7)*(1-1/11)^2/(1-2/11)*(1-1/13)^2/(1-2/13);

LCQqJiMiJiVROyImNjUiIiIiSSNDMkc2IkYnRic=

Cf1f2:=%/(1-1/2)^2/(1-1/3)^2/(1-1/5)^2/(1-1/7)^2/(1-1/11)^2/(1-1/13)^2;

LCQqJiMiJGskIiIqIiIiSSNDMkc2IkYnRic=

%*%%%; subs(C2=0.66016,%);

LCQqJiQiK3ppPVdDISIpIiIiSSNDMkc2IkYnRic=

JCIrOVNiODshIik=

sieve:=proc(N::posint) local n,B,i,j; n:=floor((N-1)/2); B:=Array(0..n-1); for j from 0 to n-1 do B[j]:=1 od; j:=0; while j<n do while B[j]=0 do j:=j+1 od; i:=2*j^2+6*j+3; if i>=n then break fi; while i<n do B[i]:=0; i:=i+2*j+3 od; j:=j+1; od; B; end;

Zio2IydJIk5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEc2JkkibkdGJkkiQkdGJkkiaUdGJkkiakdGJkYmRiZDKD5GKi1JJmZsb29yR0YmNiMsJiomIyIiIiIiI0Y2RiVGNkY2RjUhIiI+RistSSZBcnJheUdGKDYjOyIiISwmRipGNkY2Rjg/KEYtRj5GNkY/SSV0cnVlR0YoPiZGKzYjRi1GNj5GLUY+PyhGJkY2RjZGJjJGLUYqQyc/KEYmRjZGNkYmL0ZDRj4+Ri0sJkYtRjZGNkY2PkYsLCgqJkY3RjYpRi1GN0Y2RjYqJiIiJ0Y2Ri1GNkY2IiIkRjZAJDFGKkYsWz8oRiZGNkY2RiYyRixGKkMkPiZGKzYjRixGPj5GLCwoRixGNiomRjdGNkYtRjZGNkZTRjZGS0YrRiZGJkYm

debug(sieve); sieve(21);

SSZzaWV2ZUc2Ig==

{--> enter sieve, args = 21

IiM1

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKkdNWGMi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIi

IiIh

IiIk

IiIh

IiIn

IiIh

IiIq

IiIh

IiM3

IiIi

IiM2

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKkdNWGMi

<-- exit sieve (now at top level) = Array(0..9, {(1) = 1, (2) = 1, (3) = 1, (4) = 0, (5) = 1, (6) = 1, (7) = 0, (8) = 1, (9) = 1})}

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKkdNWGMi

undebug(sieve); sieve(10000);

SSZzaWV2ZUc2Ig==

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

# # Calculation of the greatest common # divisor by the Euclidean algorithm. # eucgcd:=proc(x::integer,y::integer) local u,v,r; u:=abs(x); v:=abs(y); while v<>0 do r:=irem(u,v); u:=v; v:=r od; u end;

Zio2JCdJInhHNiJJKGludGVnZXJHJSpwcm90ZWN0ZWRHJ0kieUdGJkYnNiVJInVHRiZJInZHRiZJInJHRiZGJkYmQyY+RiwtSSRhYnNHRig2I0YlPkYtLUYyNiNGKj8oRiYiIiJGOEYmMEYtIiIhQyU+Ri4tSSVpcmVtR0YoNiRGLEYtPkYsRi0+Ri1GLkYsRiZGJkYm

modinvdiv:=proc(a::integer,m::integer) local x1,x2,x3,d1,d2,d3,q,p; x1:=1; d1:=a; x2:=0; d2:=m; p:=0; do if d2=0 then if p=0 then return [x1,d1] elif x1=0 then return [x1,d1] else return [m-x1,d1] fi; fi; q:=iquo(d1,d2); d3:=d1-q*d2; x3:=x1+q*x2; p:=1-p; x1:=x2; x2:=x3; d1:=d2; d2:=d3; od; end;

Zio2JCdJImFHNiJJKGludGVnZXJHJSpwcm90ZWN0ZWRHJ0kibUdGJkYnNipJI3gxR0YmSSN4MkdGJkkjeDNHRiZJI2QxR0YmSSNkMkdGJkkjZDNHRiZJInFHRiZJInBHRiZGJkYmQyg+RiwiIiI+Ri9GJT5GLSIiIT5GMEYqPkYzRjk/KEYmRjZGNkYmSSV0cnVlR0YoQytAJC9GMEY5QCcvRjNGOU83JEYsRi8vRixGOUZDTzckLCZGKkY2RiwhIiJGLz5GMi1JJWlxdW9HRig2JEYvRjA+RjEsJkYvRjYqJkYyRjZGMEY2Rkk+Ri4sJkYsRjYqJkYyRjZGLUY2RjY+RjMsJkY2RjZGM0ZJPkYsRi0+Ri1GLj5GL0YwPkYwRjFGJkYmRiY=

modinvdiv(13874,15543);

NyQiJS4qKSIiIg==

# # Calculation of the greatest common # divisor by the binary algorithm. # bingcd:=proc(x::integer,y::integer) local u,v,k,t; u:=abs(x); v:=abs(y); if u=0 then RETURN(v) fi; if v=0 then RETURN(u) fi; k:=0; while type(u,even) and type(v,even) do k:=k+1; u:=u/2; v:=v/2 od; if type(u,odd) then t:=-v else t:=u fi; while t<>0 do while type(t,even) do t:=t/2 od; if t>0 then u:=t else v:=-t fi; t:=u-v; od; u*2^k end;

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

oddmodinvbin:=proc(a::nonnegint,m::posint) local x1,x2,x3,d1,d2,d3,p; if not type(m,odd) then return FAIL fi; if m=1 then return [0,1] fi; x1:=1; d1:=a mod m; x2:=m; d2:=m; if type(d1,even) then x3:=0; d3:=m; p:=1 else x3:=1; d3:=d1; p:=0 fi; while d3<>0 do while type(d3,even) do d3:=d3/2; if type(x3,even) then x3:=x3/2 else x3:=(x3+m)/2 fi; od; if p=0 then x1:=x3; d1:=d3 else x2:=m-x3; d2:=d3 fi; if x1>=x2 then x3:=x1-x2 else x3:=m+x1-x2 fi; if d1>=d2 then d3:=d1-d2; p:=0 else d3:=d2-d1; p:=1 fi; od; [x1,d1] end;

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

oddmodinvbin(13874,15543);

NyQiJS4qKSIiIg==

# # This procedure calculate the sum of the reciprocal # of primes up to x and compare with ln(ln(x)). # sumprimerec:=proc(x) local s,p,i; s:=0.; p:=2; while p<x do s:=evalf(s+1/p); p:=nextprime(p) od; [s,evalf(s-ln(ln(x)))] end;

Zio2I0kieEc2IjYlSSJzR0YlSSJwR0YlSSJpR0YlRiVGJUMmPkYnJCIiIUYtPkYoIiIjPyhGJSIiIkYxRiUyRihGJEMkPkYnLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMsJkYnRjEqJEYoISIiRjE+RigtSSpuZXh0cHJpbWVHRiU2I0YoNyRGJy1GNjYjLCZGJ0YxLUkjbG5HRiU2Iy1GRUYjRjtGJUYlRiU=

sumprimerec(10); sumprimerec(100); sumprimerec(1000); sumprimerec(10000); sumprimerec(100000); sumprimerec(1000000);

NyQkIit3Lz53NiEiKiQiKzIuZUBNISM1

NyQkIisucyJHIT0hIiokIip4dmp2I0Yl

NyQkIitKLDMpPiMhIiokIiooUk5hRUYl

NyQkIitlKmZJWyMhIiokIipfSnRpI0Yl

NyQkIit5QEYwRiEiKiQiKkA9IT1FRiU=

NyQkIitTIkd0KUchIiokIipEaWBoI0Yl

# # This procedure calculate the factor qsAB. # qsAB:=proc(s::posint,A::posint,B::posint) local P,p; P:=1.; p:=nextprime(A-1); while p<B do P:=P*(1-s/p); p:=nextprime(p) od; P end;

Zio2JSdJInNHNiJJJ3Bvc2ludEclKnByb3RlY3RlZEcnSSJBR0YmRicnSSJCR0YmRic2JEkiUEdGJkkicEdGJkYmRiZDJj5GLiQiIiIiIiE+Ri8tSSpuZXh0cHJpbWVHNiRGKEkoX3N5c2xpYkdGJjYjLCZGKkYzRjMhIiI/KEYmRjNGM0YmMkYvRixDJD5GLiomRi5GMywmRjNGMyomRiVGM0YvRjxGPEYzPkYvLUY3NiNGL0YuRiZGJkYm

qsAB(1,1,100);

JCIrMEg8LjchIzU=

B:=10: qsAB(1,1,B); B:=100: qsAB(1,1,B); B:=1000: qsAB(1,1,B); B:=10000: qsAB(1,1,B); B:=100000: qsAB(1,1,B); B:=1000000: qsAB(1,1,B);

JCIrJkc5ZEcjISM1

JCIrMEg8LjchIzU=

JCIrXGpfJzQpISM2

JCIrUSNwJSkzJyEjNg==

JCIrZDxIdlshIzY=

JCIrKCo0I1ExJSEjNg==

qsAB(2,7,1000000);

JCIrbHNZIT0jISM2

%*(ln(1000000.)/ln(44000.*2^25))^2;

JCIrZ1RqK2AhIzc=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn