(Unsolved) In this infinite sequence, no term is a prime: prove/disprove.

Solution 1:

We have $343434 = 2 \times 3 \times 7 \times 13 \times 17 \times 37$. $34343$ is not divisible by any of these numbers. Therefore the subsequence $34343$, $34343 + 343434 \times 10^5$, $34343 + 343434 \times 10^5 \times (10^0 + 10^6)$, $34343 + 343434 \times 10^5 \times (10^0 + 10^6 + 10^{12})$, $34343 + 343434 \times 10^5 \times (10^0 + 10^6 + 10^{12} + 10^{18})$ etc. consists of numbers which are not divisible by $2, 3, 5, 7, 13, 17, \quad \text{or}\quad 37$.

Heuristically, the chance of a random number $N$ being prime is $1 / \ln N$. Having $7$ small primes excluded as possible factors increases the chances by a factor $(2/1)(3/2)(5/4)(7/6)(13/12)(17/16)(37/36) ≈ 5.1757$.

The numbers are around $3.4343 \times 10^{5+6k}$ with $k = 0, 1, 2, 3,\ldots$ The natural logarithm is about $13.8155k + 10.4442$. So the chance that each of the numbers is a prime is about $5.1757 / (13.8155k + 10.4442)$. The expected number of primes among the numbers for $k = 0$ to $n$ is about $0.3746 \cdot\ln (n) + 0.4013$. For $n = 1,666$ the expected number of primes is about $3.1803$; that's the range that Uncountable checked. So finding no primes is slightly unlucky, but not that unlikely.

For a $50\%$ chance of finding a prime, the expected number of primes needs to increase by $\ln 2 ≈ 0.6931$, so $\ln n$ needs to be increased by $0.6931/0.3746 ≈ 1.8502$, $n$ needs to be multiplied by $6.36$. So there's a $50\%$ chance of finding a prime with up to $63,600$ digits; then a $50\%$ chance for a prime with up to $404,000$ digits and so on.

Of course all that is just heuristic. If it is correct, then a prime will almost certainly exist. Finding a probable prime might be very hard. If checking a range that gives a $50\%$ chance fails, the next range giving a $50\%$ chance is $6.36$ times larger. Miller-Rabin test grows more than quadratic with the number of digits, and there are more numbers to test, so the next range takes more than $6.36^3 = 257$ times longer.