```Easily formulated, and superficially similar questions
on prime numbers, can in fact range from the very easy
to the extraordinarily difficult, quite unexpectedly.

As an example, arrange the positive integers into four
columns, as follows:

1     2*    3*    4
5*    6     7*    8
9    10    11*   12
13*   14    15    16
17*   18    19*   20
21    22    23*   24
25    26    27    28
29*   39    31*   32
33    34    35    36
37*   38    39    40
41*   42    43*   44
45    46    47*   48
49    50    51    52
53*   54    55    56
57    58    59*   60
61*   62    63    64
65    66    67*   68
69    70    71*   72
73*   74    75    76
77    78    79*   80
81    82    83*   84
85    86    87    88
89*   90    91    92
93    94    95    96
97*   98    99   100
:     :     :     :
:     :     :     :

A primary school child can undestand why there is just
one prime in column 2, and no primes in column 4.

A secondary school student can understand Euclid's proof
that columns 1 and 3, taken together, contain infinitely
many primes.

An undergraduate student can understand why column 1
contains infinitely many primes (a theorem proved
in this course), or why there must be infinitely
many ROWS with NO primes.

A postgraduate student can understand why `half' of the
primes are in column 1 and `half' in column 3 (in a
the sense that can be made very precise).

Many mathematicians believe that there are infinitely
many ROWS containing TWO primes, but nobody has been
able to prove it.
```