While looking at a paper by Zivkovic on 0-1 matrices
http://arxiv.org/PS_cache/math/pdf/0511/0511636v1.pdf
which uses SNF (Smith normal form) to aid in computing determinants
and do other classifications, I saw that "interesting" examples had
determinants whose absolute values were not squarefree integers.
("Interesting" here means the last two components of the SNF are d
and f*d for integers d and f with d 1 , f 0, so the determinant
is divisible by d*f*d.) This led me to consider positive integers
which were multiples of a square of a prime (S), and positive
integers
that were not (F) .

It is clear that F has no sequence of consecutive integers that has
more than 3 terms. Are there infinitely many such 3 term sequences?
Are there infinitely many such with 2 terms? (A conjecture of
Schinzel and other weaker conjectures imply that yes, there are
infinitely many such pairs of squarefree consecutive numbers.)

Using the Chinese Remainder Theorem, one should (I think) be able
to construct arbitrarily long consecutive sequences from S. For n = 2
and 3, I find the smallest being (8,9) and (48,49,50). For a given n
how small can a consecutive sequence of n members be to all belong to
S?

How easily can the density of F (and so of S) be computed?

The real question is what literature is there on these questions about
S and F, and what other questions are out there?

Gerhard Paseman, 2008.07.04

In article , wrote:

While looking at a paper by Zivkovic on 0-1 matrices
http://arxiv.org/PS_cache/math/pdf/0511/0511636v1.pdf
which uses SNF (Smith normal form) to aid in computing determinants
and do other classifications, I saw that "interesting" examples had
determinants whose absolute values were not squarefree integers.
("Interesting" here means the last two components of the SNF are d
and f*d for integers d and f with d 1 , f 0, so the determinant
is divisible by d*f*d.) This led me to consider positive integers
which were multiples of a square of a prime (S), and positive
integers
that were not (F) .

It is clear that F has no sequence of consecutive integers that has
more than 3 terms. Are there infinitely many such 3 term sequences?
Are there infinitely many such with 2 terms? (A conjecture of
Schinzel and other weaker conjectures imply that yes, there are
infinitely many such pairs of squarefree consecutive numbers.)

Using the Chinese Remainder Theorem, one should (I think) be able
to construct arbitrarily long consecutive sequences from S. For n = 2
and 3, I find the smallest being (8,9) and (48,49,50). For a given n
how small can a consecutive sequence of n members be to all belong to
S?

How easily can the density of F (and so of S) be computed?

The real question is what literature is there on these questions about
S and F, and what other questions are out there?

I think most of the questions here are unsolved. The density
of the squarefree numbers is an exception - there are simple proofs
that that density is 6-over-pi-squared. I'm sorry that I can't give you
up-to-date references on these problems, but the keyword is
squarefree, and Guy's Unsolved Problems In Number Theory (3rd
edition) is probably a good place to start.

--
Gerry Myerson ) (i - u for email)