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Prime power puzzle (Posted on 2017-04-01) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Math Man    
Rating: 5.0000 (1 votes)

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Half A proof | Comment 4 of 8 |
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
  Posted by Steve Herman on 2017-04-02 11:17:51

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