Construct a grid six spaces wide and with as many rows as you like. Number the spaces 1, 2, 3,…. Highlight the spaces containing primes. Why, after the first row, do primes only appear in spaces in the first or fifth columns? I only went up to 503; is there a proof this (one hesitates to call it a) pattern continues, if it does?
prime numbers (except 2 and 3) cannot be divisible by 2 or 3…
so they will be of the form 6n +1 or 6n + 5
I mean, I understand why there can’t be any in the second, third, fourth, or sixth columns. So I guess that’s the answer: because where the f else would they go? But something about it seems odd to me.
Yep except for 2 in the first row, they’re all odd. 
Of course, 2 is the only even prime, which makes it the oddest.
If you think that’s odd, take a look at the Ulam Spiral.
If you think that’s odd, take a look at the Junji Ito Spiral.
This might interest you.
For any number of columns, the columns that have a first-row number that is multiple of one of the prime factors of the total number of columns cannot contain a prime. This is because for each number, you add the number of columns to get the number in the next row.
Using 6 columns: In column 2, for example, if you add 6, which is a multiple of 2, you will get another multiple of 2. In column 3, you are adding 6 which is a multiple of 3, so the result will always be a multiple of 3. The last column will always have a multiple of the number of columns, so it cannot contain a prime. Similar for columns 4 and 6. That leaves columns 1 and 5 to contain all the primes.
Try 2 x 3 x 5 = 30 columns. The following columns cannot contain primes: 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30. All the even-numbered columns will always contain even numbers, since you are adding an even number to get the next row. Column 21 won’t contain primes because it is a multiple of 3 and you are always adding a multiple of 3. Column 25 will always contain multiples of 5.
When you use a prime number as the number of columns, the only column without primes is the last column because there are no prime factors lower than that number.
As others have noted, it’s easy to prove that the primes can only appear in columns 1 and 5.
What’s deeper and not so easy to prove is that columns1 and 5 have approximately equal densities of primes, as a consequence of Dirichlet’s Theorem on arithmetic progressions.