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)





