at most $ 4$ steps to convert from one prime $a$ to another prime $b$ if the intermediate number formed in the process is also a prime . We have a prime number $a$ and we want to convert it into another prime number $b$   but we can either add a prime number to it or subtract a prime number from it and the intermediate number formed in the process should also be a prime   .
The solution says that if there exists some path such that we can do this , then the at most number of paths is 4 if we want it to done with minimum number of operations of addition and subtraction ? Can anyone prove this statement ?
eg if $a$ is $5$ and $b$ is $43$ then a path is $5$ >$2$>  $43$ i.e ($5$-$2$=$3$ is a prime and $43$-$2$=$41$ is a prime).Here path is of length $3$
 A: To get from $29$ to $41$ the shortest path is
$$
29 \to 31 \to 2 \to 43 \to 41.
$$
To make sense of the question, I think you have to interpret the "length" of
the path as the number of times the symbol $\to$ occurs between
two primes in the path, not the number of primes in the path.
To show that there is no longer path, consider that to "move"
from an odd prime to any other prime, the only possibilities
are to add or subtract $2$ in order to reach another odd prime,
or to subtract another odd prime to reach $2$.
Moreover, there is never a case where you can add $2$ twice in a row,
that is, never a case where $p$, $p+2$, and $p+4$ are all prime.
So let's make a connection graph with prime numbers as nodes and
possible "moves" as edges.
There are edges from the prime number $2$ to many other prime numbers,
but from each of those nodes there is at most one other edge.
The graph consists of $2$, 
some odd primes that are connected to no other odd primes,
and some pairs of odd primes $p, p+2$ that are connected to each other.
Some of the odd primes are connected to $2$, and there are no other edges.
There are not very many different cases to consider for what could be
the shortest path between two primes $p$ and $q$ if any
path between those nodes exists.
