interlaced primes

[Mini-Geek]

Originally Posted by davar55

If one were to ask How Many 100- 200- or 300- digit numbers
were interlaced primes, would that be too hard to compute?

I'm going to make a guess here and say: yes, it's too hard. It could certainly be estimated to an accurate degree, but not computed precisely.

Originally Posted by davar55

What n- digit value would be not too hard to count?

My slightly-educated guess based on http://primes.utm.edu/nthprime/ being calculated no later than 2004 to (as you'd describe it) n=13:
n=20 would take enough effort to consider it a (soft) maximum.

[science_man_88]

Originally Posted by Mini-Geek

I'm going to make a guess here and say: yes, it's too hard. It could certainly be estimated to an accurate degree, but not computed precisely.

My slightly-educated guess based on http://primes.utm.edu/nthprime/ being calculated no later than 2004 to (as you'd describe it) n=13:
n=20 would take enough effort to consider it a (soft) maximum.

I like your opinion I tried in pari but davar already got a message from me with it:

a=0;until(a>10^99-1 && a<10^100 && (((a%100)-(a%10))/10)%2==1 && isprime(a),a=randomprime(10^100));a

I checked isprime because randomprime() 's description says strong pseudoprime.

[VBCurtis]

Originally Posted by davar55

If one were to ask How Many 100- 200- or 300- digit numbers
were interlaced primes, would that be too hard to compute?
What n- digit value would be not too hard to count?

Let's apply the Prime Number Theorem for the 100 digit case:
There are (very roughly) 4e96 100-digit primes. Each of the 50-digit interlaced numbers is prime 1/log(10^49) of the time. Since both of the interlaced numbers need to be prime, we square that probability to see how often a 100-digit number is 2-interlaced prime: roughly 1 in 13000.

4e96 / 13000 = 3e92, again very roughly.

So, there are quite a few such 100-digit primes.

[davar55]

OK Nice work everyone.

How about this extension of the original puzzle:

For each Mersenne Prime M concatenate
the smallest positive integer N (no lead zeroes)
to it that forms a prime P = M,N (comma meaning concatenate)
which is 2-interlaced? If that's too easy, 2-and 3- interlaced?

Since proving these prime or even prp gets too hard too fast
(you guys know better)
how does the sequence of such N begin, and how high can we go,
now or possibly ever? (Two separate sequences, for 2- and 3- ).