Problems & Puzzles: Problems

Problem 8.- (m, n+1) and (m+1, n) with the same set of prime factors.

Motzkin and Strauss asked for all pair of numbers (m, n) such that (m, n+1) have the same set of distinct prime factors, and similarly for (m+1, n). It was thought that such pairs were necessarily of the form m = 2 k+1 and n = m 2 -1, (k = 0, 1, 2, 3, …), until Conway found the following counterexample :

For (m, n) = (35, 4374), (m, n+1) = (5 . 7, 5 4 . 7), as required and (m+1, n) = (2 2 . 3 2, 2 . 3 7 ), as required also, but obviously m = 35 is not of the form 2 k + 1 ; neither n = m 2 -1

Can you find other counterexamples ?

(Ref. 2, p. 75, B19)


 


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