Puzzle 1220 Prime Split

Puzzle 1220 Prime Split

Giorgos Kalogeropoulos sent the following puzzle:

Split k into two primes p1 and p2 such that the largest prime factor (LPF) of k+1 is p1 and LPF(k-1) = p2.

The smallest k with this property is 323.

We split k = 323 into two primes 3-23, so p1 = 3 and p2 = 23.

LPF(k+1) = LPF(324) = LPF(2^2*3^4) = 3 = p1

LPF(k-1) = LPF(322) = LPF(2*7*23) = 23 = p2

Next k with this property is 357149993.

We split k into two primes 3571-49993, so p1 = 3571 and p2 = 49993

LPF(357149994) = LPF(2*3*79*211*3571) = 3571 = p1

LPF(357149992) = LPF(2^3*19*47*49993) = 49993 = p2

It is allowed for p2 to have leading zeros.

Q1. Find the first five integers with this property
Q2. What is the biggest k that you can find with this property?

Q3. Can you find at least one prime number k with this property?