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## Main Question or Discussion Point

If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors.

For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5).

Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which is 5. But where is the last occurrence of 5 as the maximum prime factor between two neighboring integers? Is there a known way to compute this, or to get a bound on it?

Essentially what I am looking for is a function that take a prime and tells me which integer n is part of the last pair of integers that have that prime as the greatest prime factor.

n_last = f(p)

For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5).

Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which is 5. But where is the last occurrence of 5 as the maximum prime factor between two neighboring integers? Is there a known way to compute this, or to get a bound on it?

Essentially what I am looking for is a function that take a prime and tells me which integer n is part of the last pair of integers that have that prime as the greatest prime factor.

n_last = f(p)