## Prime connection

### Problem 425

Published on Saturday, 27th April 2013, 04:00 pm; Solved by 729; Difficulty rating: 25%
Two positive numbers A and B are said to be *connected* (denoted by "A ↔ B") if one of these conditions holds:

(1) A and B have the same length and differ in exactly one digit; for example, 123 ↔ 173.

(2) Adding one digit to the left of A (or B) makes B (or A); for example, 23 ↔ 223 and 123 ↔ 23.

We call a prime P a *2's relative* if there exists a chain of connected primes between 2 and P and no prime in the chain exceeds P.

For example, 127 is a 2's relative. One of the possible chains is shown below:

2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127

However, 11 and 103 are not 2's relatives.

Let F(N) be the sum of the primes ≤ N which are not 2's relatives.

We can verify that F(10^{3}) = 431 and F(10^{4}) = 78728.

Find F(10^{7}).