Thank you Jan = Frank_King
, excellent news!
The numbers with decimal representation like 91, 991, 9991, 99991, ... are visually striking, and it's nice that we can say that those with an odd number of digits are "potentially" primes.
Of course, when a number N
is composite, an easy proof is to display two smaller numbers with product equal to N
. However, when N
is prime, the proof that this is so can be much more demanding. As you correctly and cautiously point out, probabilistic methods have been developed that are capable of telling us, in a "reasonable" amount of time, that N
is (probably) prime. When the consequences of this being wrong are not too important, that is a satisfactory state of knowledge, and in the present case your report about the 33 digit and 45 digit numbers lends support to my conjecture that there are infinitely many primes with the 99...91 pattern. On the other hand, if a particular "probable prime" N
turning out to be composite will risk a bridge collapsing or national security being breached, then that is a much higher order of seriousness... Of course, ultimately there are tests which in principle can determine definitively whether a given N
is prime, but the time taken to complete such a test might be impractically long (greater than the expected lifetime of our Sun, for example). So, for practical reasons, we might have to be content with probabilistic tests and a conclusion that N
is (probably) prime.
I'm sure you know all this Jan = Frank_King
, but it seems worthwhile to say it for others who might not be familiar with such ideas...
Inspired by the report from Jan = Frank_King
, I used a probabilistic prime test (perhaps the same test as Jan used) to verify that the k
digit number N_k
= 99..91 is (probably) prime when k
= 33 and k
= 45, and there are no further examples with 9 ≤ k
≤ 99. However, the test shows that, in fact, the number with k = 105
is (probably) prime!
And that's not all! The range 9 ≤ k
≤ 225 has FIVE examples of (probable) primes:
k = 33
k = 45
k = 105
k = 197
k = 199
The last two are not twins in the usual sense, but they are consecutive members in the sequence of numbers we are considering
This continues to support my conjecture, but it's good to remember that the larger k
is, the greater the chance that the probabilistic test for primality fails to identify that N_k
is actually composite...