mathematics - Standard Sudoku Specialized with just 2 Primes

A standard 9x9 Sudoku uses the digits 1 to 9.

You are only allowed two distinct primes to represent 1 to 9.

Find out the minimum number of characters (digits + signs) needed to construct a Sudoku puzzle with a unique solution.

Allowed signs are plus, minus, division, multiplication, factorial, and exponentiation.

Answer

The minimal Sudoku contains 17 digits: 2 each of seven of the nine digits, 3 of another digit, and none of the last one. (By symmetry, it doesn't matter which two digits are selected to appear 3 and 0 times in the Sudoku.)

If we select our two primes as

2 and 3, then we can represent the digits as

$$3-2,2,3,2+2,2+3,3!,?,2\wedge3,3\times3.$$ (We omit 7 because we only need to represent all but one of the digits for the minimal Sudoku.)

So we choose the minimal Sudoku with

seventeen digits filled, two each of $1,2,4,5,6,8,9$ and three of $3$.

How many symbols do we need to use then?

Each of $1,4,5,8,9$ requires three symbols, $2$ and $3$ require only one, and $6$ requires two. So the total is

$$5(2\times3)+(2\times1)+(2\times2)+(3\times1)=30+2+4+3=39.$$