603, 604, and 605 are the first 3 consecutive integers that are the product of a prime and another prime squared.
603=32*67
604=22*151
605=5*112
1. What is the first set of 4 consecutive integers that are the product of a prime and another prime squared?
2. What is the first set of 5 consecutive integers that are the product of a prime and another prime squared?
I have looked a Charlie's suggested approach to proving that 4 in a row is not possible.
I can proved that one cannot have 4 consecutive integers of the form [m,4p,n,2q^2] or [4p,n,2q^2,m] where p and q are prime. This would lead to 4p = 2q^2 -2
which simplifies to 2p = (q-1)(q+1) which has no prime solutions.
Thus, 6 in a row is impossible.
4 in a row would also be impossible if we could prove that one cannot have 4 consecutive integers of the form [n,2q^2,m,4p] or [2q^2,m,4p,n] where p and q are prime. This leads to 2p = q^2+1, which is obviously not impossible.
Solutions include (p,q) = (5,3), (13,5), and (61,11)
So, I have not yet proved that 4 is impossible
Edited on April 2, 2017, 1:57 pm
Half A proof
