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+K1}
B = {b, b+1, b+2,..., b+K1}
b>a+K1
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
