Problem Statement

As used here, all unprimeable numbers   (positive integers)   are always expressed in base ten.

───── Definition from OEIS ─────: Unprimeable numbers are composite numbers that always remain composite when a single decimal digit of the number is changed.

───── Definition from Wiktionary   (referenced from Adam Spencer's book) ─────: (arithmetic)   that cannot be turned into a prime number by changing just one of its digits to any other digit.   (sic)

Unprimeable numbers are also spelled:   unprimable. All one─ and two─digit numbers can be turned into primes by changing a single decimal digit.

190   isn't unprimeable,   because by changing the zero digit into a three yields   193,   which is a prime.

The number   200   is unprimeable,   since none of the numbers   201, 202, 203, ··· 209   are prime, and all the other numbers obtained by changing a single digit to produce   100, 300, 400, ··· 900,   or   210, 220, 230, ··· 290   which are all even.

It is valid to change   189   into   089   by changing the   1   (one)   into a   0   (zero),   which then the leading zero can be removed,   and then treated as if the   "new"   number is   89.

Show all output here, on this page.

Let's start with the solution:

bool is_unprimeable(int i)
{
    string s = i->digits();
    for(int offset; offset < sizeof(s); offset++) {
	foreach("0123456789"/1, string repl) {
	    array chars = s/1;
	    chars[offset] = repl;
	    int testme = (int)(chars*"");
	    if( testme->probably_prime_p() )
		return false;
	}
    }
    return true;
}

void main()
{
    int i, count;
    array unprimes = ({});
    mapping first_enders = ([]); // first unprimeable ending with each digit 
    while(sizeof(first_enders) != 10) {
	i++;
	if( is_unprimeable(i) ) {
	    count++;
	    unprimes += ({ i });
	    string last_digit = i->digits()[<0..];
	    if( !first_enders[last_digit] )
		first_enders[last_digit] = i;
	}
	werror("%d\r", i); // Progress output
    }
    
    write("First 35 unprimeables: %s\n\n", (array(string))unprimes[0..34]*" ");
    write("The 600th unprimeable is %d\n\n", unprimes[599]);
    write("The first unprimeable number that ends in\n");
    foreach(sort(indices(first_enders)), string e) {
	write("  %s is: %9d\n", e, first_enders[e]);
    }
}