ON ANOTHER ERDÖS’ OPEN PROBLEM Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA E-mail: smarand@unm.edu Paul Erdös has proposed the following problem: (1) “Is it true that lim max(m d(m)) n ?, where d(m) represents the n
m n
number of all positive divisors of m .” We clearly have : Lemma 1. ( )n N \ 0,1, 2 , ( )! s N* , ( )! 1 ,...,
s
N,
s
0 , such that
n p1 1 ps s 1 , where p1, p2 ,... constitute the increasing sequence of all positive primes. * ps s 1 , where Lemma 2. Let s . We define the subsequence ns (i) p1 1 , such that s 0 and 1 ... and we 1 ,..., s are arbitrary elements of s order it such that ns (1) ns (2) ... (increasing sequence). We find an infinite number of subsequences ns (i) , when s traverses * , with the properties: * a) lim ns (i) for all s . i
b) ns1 (i), i N*
ns2 ( j ), j N*
c) N \ 0,1, 2
ns (i), i N*
, for s1
s2 (distinct subsequences).
s N*
Then: Lemma 3. If in (1) we calculate the limit for each subsequence
ns (i) we
obtain:
lim n
max (m d (m)) m p1 1 ps s
lim (
n
1
1)...(
s
p1 1
ps s 1
1) 1
lim
n
1
lim p1 1 n
...
ps s
(
s
From these lemmas it results the following: Theorem: We have lim max(m d(m)) n n
m n
1
.
1
1)...(
s
1) p1 1
ps s 1