Problems & Puzzles: Puzzles

Puzzle 86.- Disjoint sets A & B of consecutive numbers sharing the same set P of prime divisors

Let A & B to be two sets each of one of K consecutive numbers, such that:
a) A & B do not share any member
b) A & B share the same set P of prime divisors

Example:

for K=3: A={3, 4, 5} and B={8, 9, 10} are the least disjoint sets of consecutive numbers that share the same set P={2, 3, 5} of prime divisors.

Ėrdos conjectures that there are only finite examples of these couples of sets. See: B35, p. 90, R. K. Guy.

Here are a few results of mine after a very little and preliminary search:

A = {a, a+1, a+2,..., a+K-1}
B = {b, b+1, b+2,..., b+K-1}
b>a+K-1
P = {p1, p2, p3,..., pn}

 K a b Shared set of primes Comment 3 3 8 2 3 5 The least 3 4 8 2 3 5 3 5 14 2 3 5 7 3 5 48 2 3 5 7 3 12 26 2 3 7 13 3 13 63 2 3 5 7 13 3 14 48 2 3 5 7 3 20 54 2 3 5 7 11 3 20 98 2 3 5 7 11 3 33 119 2 3 5 7 11 17 3 34 49 2 3 5 7 17 3 54 98 2 3 5 7 11 3 55 75 2 3 5 7 11 19 3 89 623 2 3 5 7 13 89 3 169 323 2 3 5 13 17 19 3 2650 58563 2, 3, 5, 11, 13, 17, 53, 241 The largest known example; by Jud McCranie, 18/03/2000 4 12 25 2 3 5 7 13 The least 4 13 25 2 3 5 7 13 4 14 48 2 3 5 7 17 4 19 54 2 3 5 7 11 19 4 152 340 2 3 5 7 11 17 19 31 ? ? ? ??? 5 12 24 2 3 5 7 13 The least 5 13 48 2 3 5 7 13 17 ? ? ? ?

Questions:

a) Can you extend this table for each K<6?
b) Can you find one example for K=>6
c) Jud McCranie suggests to ask for the sets with largest quantity of members in the set P, for each K value.

Solution

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