I thought this was interesting...

starting with 23 and appending 3 after 3 makes a number that that is coprime to all primes less than 17.

it doesn't divide 2...

And I have a conjecture that if the number has a prime number of digits it has a good chance of being prime, but is sometimes also prime for even number of digits.

2333.... mod 3 is always 2.

it doesn't divide 5 because all multiples of 5 end in 0 or 5.

Now looking at 2333... mod 7:

23 mod 7 = 2

233 mod 7 = 2

and by induction on the division algorithm it must always be 2.

23 mod 11 = 1

233 mod 11 = 2

2333 mod 11 = 1

23333 mod 11 = 2

and also this pattern must repeat

23 mod 13 = 10

233 mod 13 = 12

2333 mod 13 = 11

23333 mod 13 = 6

233333 mod 13 = 9

2333333 mod 13 = 2

23333333 mod 13 = 10

and then repeats

mod 17 is interesting...

23333333 mod 17 is 0 so it is not prime, then it cycles through every remainder possible with 17 in a patter and

233333333333333333333333 (16 more 3's) is the next number that divides 17 evenly... and so on

Now factoring shows that:

233333 = 353*661

and 233333333 divides 29. adding 28 more 3's on the end again divides 29.

I thought maybe if there were a prime number of digits the number was prime but no. This type of number with 11, 17, and 23 and 54 digits are all prime, but 7, 13 and 19 are not. It does seem non-prime for every composite though.

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