Sieve of Eratosthenes

The sieve of Eratosthenes is the classical method of generating prime numbers. The
procedure is very easy to implement in Mathematica. We make a list of integers
and remove the multiples of two, of three, and so on, until all we are left with are
primes. If we generate a table of primes up to n, then it is necessary to eliminate
the multiples of the primes up to square root of n.

In[24]:=

  SieveOfEratosthenes[n_]:= 
           Module[ {t,P,i=1,x},
           t=N[Sqrt[n]];
           P=Range[2,n];
           While[ P[[i]] <= t,
             
            x=Table[k P[[i]],
                       {k,P[[i]], Floor[n/P[[i]]]}];
                        
            P=Complement[P,x];
             i++];
            Return[P]]

In[25]:=

  SieveOfEratosthenes[100]

Out[25]=

  {2, 3, 5, 7, 11, 13, 17, 19, 23, 
    29, 31, 37, 41, 43, 47, 53, 59, 
    61, 67, 71, 73, 79, 83, 89, 97}

Remarks: Do not use this sieve to generate primes for n more than 10^5 as it is
likely to be very slow. For larger n, it is better to modify the sieve to generate
primes between n1 and n2, by reading in a list of primes up to square root of n2.

Also, it is better to save the result in a file rather than display it on a screen. This
can be done by the >> operator and the file can be read in using << or the
ReadList command.

In[26]:=

  SieveOfEratosthenes[ 1000]>> primes;

In[27]:=

  
  primes= ReadList[ "primes", Expression];

In[28]:=

  primes

Out[28]=

  {{2, 3, 5, 7, 11, 13, 17, 19, 23, 
     29, 31, 37, 41, 43, 47, 53, 59, 
     61, 67, 71, 73, 79, 83, 89, 97, 
     101, 103, 107, 109, 113, 127, 
     131, 137, 139, 149, 151, 157, 
     163, 167, 173, 179, 181, 191, 
     193, 197, 199, 211, 223, 227, 
     229, 233, 239, 241, 251, 257, 
     263, 269, 271, 277, 281, 283, 
     293, 307, 311, 313, 317, 331, 
     337, 347, 349, 353, 359, 367, 
     373, 379, 383, 389, 397, 401, 
     409, 419, 421, 431, 433, 439, 
     443, 449, 457, 461, 463, 467, 
     479, 487, 491, 499, 503, 509, 
     521, 523, 541, 547, 557, 563, 
     569, 571, 577, 587, 593, 599, 
     601, 607, 613, 617, 619, 631, 
     641, 643, 647, 653, 659, 661, 
     673, 677, 683, 691, 701, 709, 
     719, 727, 733, 739, 743, 751, 
     757, 761, 769, 773, 787, 797, 
     809, 811, 821, 823, 827, 829, 
     839, 853, 857, 859, 863, 877, 
     881, 883, 887, 907, 911, 919, 
     929, 937, 941, 947, 953, 967, 
     971, 977, 983, 991, 997}}

Notice the extra set of braces around the expression. These can be removed
using the Flatten command.

In[29]:=

  primes=Flatten[primes];
  primes[[50]]

Out[29]=

  229

Exercise: Write a function Sieve[n,m] to create a list of primes between n and m. To start
the sieve you will need a list of primes up to square root of m.

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