Problems & Puzzles: Problems

Problem 45 . A follow up to Problem 8

Maybe is a good idea you to read first the Problem 8

Well, in the next paragraph of the B19 article of the UPiNT of R. K. Guy (p.75) we read this:

"Similarly, Erdos asks if there are numbers m, n (m<n) other than m=2^k-2, n=2^k(2^k-2) such that m and n have the same prime factors and similarly for m+1 and n+1."

There we read also that Makowski found the following counterexample:

m=3*5^2, n=3^5*5, m+1=2^2*19, n+1=2^6*19

Question:

Can you find 3 more counterexamples? Jud McCRanie, Bill Murphy and Frank Rubin worked without success looking for these asked counterexamples.

"no other solutions for m and n < 10^9." (Jud McCranie)

"In my search for solutions to Problem 8 I also searched for solutions to Problem 45.  While I was away visiting my grandchildren a few months back I let my computer chug away at the search all the way to 500,000,000,000.  No further counterexamples showed up.  It simply may be that each problem has only one counterexample. (Frank Rubin)

"For clarity, I'd add to your problem #8 description that m should be less than n. If there is another counter example, then m+n > 1 000 000 000 (Bill Murphy)

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