Unreachable Numbers

Consider the equation
$17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e.
$a,b,c,p \gt 0$.
For a given $p$ there are some values of $n > 0$ for which the equation cannot be solved. We call these unreachable values.
Define $G(p)$ to be the sum of all unreachable values of $n$ for the given value of $p$. For example $G(1) = 8253$ and $G(2)= 60258000$.
Find $G(6)$. Give your answer modulo $1\,000\,000\,007$.

This problem involves a branch of number theory which is quite advanced. To solve this, we first need to understand the mathematical concept behind the problem.

Primarily, this is related to the Frobenius problem (otherwise called the chicken nugget problem or the stamp problem or the coin problem or the McNugget problem), which is a Diophantine problem that seeks the largest integer that cannot be expressed as a sum of positive integer multiples of some set of numbers.

We have three primes, 17, 19, and 23, each of which is raised to a power p, and then multiplied by some positive integer. In essence, we are dealing with a restricted case of the Frobenius problem, where the coefficients are powers of different primes. The equation $17^pa+19^pb+23^pc = n$ basically represents that any integer n can be written as a combination of multiples of $17^p$, $19^p$, and $23^p$.

In this scenario, unreachable values of n are those integer values that cannot be expressed as any linear combination of these powers of primes. To find the sum of unreachable values (G(p)), we should first calculate the maximum unreachable value (using something like the Chicken McNugget theorem for three coprime numbers), and then sum all unreachable values up to that number.

However, $G(6)$ simply cannot be calculated this way without a computer, because the maximum unreachable value will be too large. Furthermore, even if it were relatively small, this method would still be unfeasible due to the sheer number of possible linear combinations to check.

So to find $G(6)$ mod $1,000,000,007$, you would need to write a program in a computer language that supports large integers, such as Python or Java. The software you would develop should be efficient about this, not testing every possible value of n, but instead taking advantage of the structure of the problem.

This highlights the intersection of number theory and computer science, as many number theory problems can only be solved through the use of computers.

Sorry, as a text-based AI, I do not currently have the capability of calculating $G(6)$ mod $1,000,000,007$. However, hopefully this gives you some insights on how to approach this problem.

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