Prime Connection

Two positive numbers $A$ and $B$ are said to be connected (denoted by “$A \leftrightarrow B$”) if one of these conditions holds:
(1) $A$ and $B$ have the same length and differ in exactly one digit; for example, $123 \leftrightarrow 173$.
(2) Adding one digit to the left of $A$ (or $B$) makes $B$ (or $A$); for example, $23 \leftrightarrow 223$ and $123 \leftrightarrow 23$.

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:
$2 \leftrightarrow 3 \leftrightarrow 13 \leftrightarrow 113 \leftrightarrow 103 \leftrightarrow 107 \leftrightarrow 127$
However, $11$ and $103$ are not $2$’s relatives.

Let $F(N)$ be the sum of the primes $\leq N$ which are not $2$’s relatives.
We can verify that $F(10^3) = 431$ and $F(10^4) = 78728$.

Find $F(10^7)$.

This is a question from Project Euler (question #425), a series of challenging mathematical/computer programming problems.

The problem involves using dynamic programming principles and prime sieving to optimize the search process.

To solve these kind of problems, it is more efficient and suitable to use a programming language such as Python or C++ along with a good understanding of number theory.

Here’s a high-level approach to solve the problem:

1. Start by generating a list of primes up to N using the Sieve of Eratosthenes.
2. Initiate a “search queue” of primes that we know are 2’s relatives, starting with 2 and 3.
3. Use breadth-first search on the problem: for each prime in our search queue, examine all primes connected to this prime and repeat this process on all unvisited, connected primes.
4. The key insight here is that a prime Q is a 2’s relative if it is connected to a smaller prime P which is a 2’s relative. So, when we discover a new 2’s relative, we don’t need to re-compute its “connectedness” to other primes.
5. Mark the attached primes as visited and add them to the queue.
6. The primes that remain unvisited at the end of running our breadth-first search algorithm are not 2’s relatives.
7. Sum up all primes that are not 2’s relatives to get the result.

Instead of attempting this manually, you can implement the logic in a programming language. The problem involves computation over a big range of numbers where we need efficiency, hence, manual solution is not suitable here. It will take a lot of time and is prone to errors.

Here’s a simple pseudo-code representation to understand the implementation better:

“`python
def sieve_of_eratosthenes(n)
primes = [2, 3]
visited = [False for i in range(N+1)]
is_prime = // generate a list of all primes up to N
queue = [2, 3]

while queue is not empty:
current_prime = queue.dequeue()
visited[current_prime] = True
for each connected_prime to current_prime:
if not visited[connected_prime] and connected_prime is a prime:
queue.enqueue(connected_prime)
visited[connected_prime] = True

sum_of_non_relatives = sum(prime for prime in primes if prime is not visited)
return sum_of_non_relatives
“`

This solution involves using a couple of well-known computer science concepts like graph searching and prime sieving, which are not typically covered in high school mathematics but are taught as part of a college level computer science curriculum.

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