Someday soon I'll have a post where figures will be useful. Until then, more New England.
, I asked for the “smallest” 3×3 singular matrix, each of whose entries is a distinct prime. By “smallest”, I am adding all the entries in the matrix.
It turns out that there is an optimal matrix, by which I mean there is a way to arrange the smallest 9 primes so that the matrix is singular, and the sum is 100. One such way (remember, we can exchange rows, exchange columns, or take transposes and still have a solution) is this:
As to how I found this solution, I used the following observation for such a solution M: The columns of M are linearly dependent over the integers. Thus I wrote a short program in MATLAB that looked through the first N primes, took them two at a time, say p and q, and then checked whether ap+bq was a prime, for parameters a and b that I could change.
Using a=1, b=-2, I found the optimal matrix is
So much New England.
whose sum is 112, and I thought this was pretty good, since it only missed 19 and 29. The next choice of a = 3, b = -2 turned out to be the winner above (notice, as expected, that 3 times the first column minus twice the second is the third). I was also surprised to find that choosing a = 5, b = -6 did very well, with a matrix total of 106 (this is, by necessity, the second best answer, since it only swaps out a 23 for a 29).
Since yesterday, I was able to use the same sort of ad-hoc method to find a similarly optimal 4×4 “prime” matrix (again, “optimal” means I use only the first 16 primes):
which is enough for me to ask the following question, verified only for the first two cases:
For n > 2, is there a singular n x n matrix whose entries are the first n^2 prime numbers?
Too much New England! Creepy woods women!