problem0():
  Which numbers are prime?

  n =  31  is prime? =  True
  n =  331  is prime? =  True
  n =  3331  is prime? =  True
  n =  33331  is prime? =  True
  n =  333331  is prime? =  True
  n =  3333331  is prime? =  True
  n =  33333331  is prime? =  True
  n =  333333331  is prime? =  False

problem1():
  Is 1,000,009 prime?

  n =  1000009  is prime? =  False

problem2():
  Check Euler's formula for primes.
  n =  0   f(n) = 41  is prime.
  n =  1   f(n) = 43  is prime.
  n =  2   f(n) = 47  is prime.
  n =  3   f(n) = 53  is prime.
  n =  4   f(n) = 61  is prime.
  n =  5   f(n) = 71  is prime.
  n =  6   f(n) = 83  is prime.
  n =  7   f(n) = 97  is prime.
  n =  8   f(n) = 113  is prime.
  n =  9   f(n) = 131  is prime.
  n =  10   f(n) = 151  is prime.
  n =  11   f(n) = 173  is prime.
  n =  12   f(n) = 197  is prime.
  n =  13   f(n) = 223  is prime.
  n =  14   f(n) = 251  is prime.
  n =  15   f(n) = 281  is prime.
  n =  16   f(n) = 313  is prime.
  n =  17   f(n) = 347  is prime.
  n =  18   f(n) = 383  is prime.
  n =  19   f(n) = 421  is prime.
  n =  20   f(n) = 461  is prime.
  n =  21   f(n) = 503  is prime.
  n =  22   f(n) = 547  is prime.
  n =  23   f(n) = 593  is prime.
  n =  24   f(n) = 641  is prime.
  n =  25   f(n) = 691  is prime.
  n =  26   f(n) = 743  is prime.
  n =  27   f(n) = 797  is prime.
  n =  28   f(n) = 853  is prime.
  n =  29   f(n) = 911  is prime.
  n =  30   f(n) = 971  is prime.
  n =  31   f(n) = 1033  is prime.
  n =  32   f(n) = 1097  is prime.
  n =  33   f(n) = 1163  is prime.
  n =  34   f(n) = 1231  is prime.
  n =  35   f(n) = 1301  is prime.
  n =  36   f(n) = 1373  is prime.
  n =  37   f(n) = 1447  is prime.
  n =  38   f(n) = 1523  is prime.
  n =  39   f(n) = 1601  is prime.

  Euler's formula fails for n =  40
  fn =  1681  is not prime.

problem3():
  Count calls to mod function for prime tests

  is_prime1( 27644437 ) =  True   calls =  27644435
  is_prime2( 27644437 ) =  True   calls =  27644435
  is_prime3( 27644437 ) =  True   calls =  5256
  is_prime4( 27644437 ) =  True   calls =  1754

problem4():
  prime factorization of n

  Factors of  1120
[2, 2, 2, 2, 2, 5, 7]

  Factors of  2023
[7, 17, 17]

  Factors of  314159265
[3, 3, 5, 7, 127, 7853]

problem5():
  prime_pi(n): number of primes up to n.

  prime_pi( 64 ) =  18
  prime_pi( 256 ) =  54
  prime_pi( 1024 ) =  172

problem6():
  Compare prime_pi(n) and n/logn.
  n =  256   prime_pi(n) =  54   gauss =  46.16624130844683
  n =  512   prime_pi(n) =  97   gauss =  82.07331788168325
  n =  1024   prime_pi(n) =  172   gauss =  147.73197218702984
  n =  2048   prime_pi(n) =  309   gauss =  268.6035857945997
  n =  4096   prime_pi(n) =  564   gauss =  492.43990729009954

problem7():
  sigma(n): sum the divisors of n.

  sigma( 617 ) =  618
  sigma( 816 ) =  2232
  sigma( 1000 ) =  2340
  sigma( 1024 ) =  2047

problem8():
  tau(n): count the divisors of n.

  tau( 512 ) =  10
  tau( 610 ) =  8
  tau( 832 ) =  14
  tau( 960 ) =  28

problem9():
  gpf(n): greatest prime factor of n.

  gpf ( 25 ) =  5
  gpf ( 698 ) =  349
  gpf ( 751 ) =  751
  gpf ( 364 ) =  13
  gpf ( 526 ) =  263

problem10():
  Multiply big numbers
  diff =  0