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
|
