Wikipedia talk:Articles for deletion/Radical integer

message: Just got a reply from Schroeppel. --Salix alba (talk) 08:17, 12 April 2006 (UTC)
replies: text: p.s. I've got other email addresses for Schroeppel, ask if needed.
I looked at the WP deletion criteria, and the discussion of the
Radical_Integer article. I don't understand the keep-or-drop criteria for Wikipedia.
MathWorld has a mistake about "cubic" - it should say "quintic".
The algebraic integer phi = (1 + sqrt5)/2 = 1.618 = golden ratio
is an alg-int because it's a root of X^2-X-1 = 0, leading coefficient
(of X^2) is 1, and other coeffs are (plain vanilla) integers.
It's obvious from the expression that it's an algebraic number, but
not obvious that it's an (algebraic) *integer* because the formula
has a /2 in it.  The algebraic number (1 + sqrt7)/2 is NOT an alg-int.
In fact, the expression generates an alg-int iff 5 is replaced with
another 4K+1 number like 13,17, or -3; 9 works also.
However, phi = cbrt(2 + sqrt5); in this form, the integer-ness is
now obvious.
It turns out that (1 + cbrt10 + cbrt100)/3 is also an alg-int, and
some playing around turned up a no-division expression for it.
I posed the question (on the math-fun list) whether
any alg-int had some expression that made the integerness obvious.
There's an obvious problem with alg-ints that don't have any
radical expression whatsoever (such as most roots of quintic equations).
So the revised question is whether an alg-int that has some radical
expression also has a radical expression without division.
I invented "radical integer" to describe these expresions.
(Technical nit for the WP article: the index of the root must be a
positive integer, since the -1 root is just the reciprocal.)
Several people on the mailing list contributed ideas and discussion.
I was able to find a proof that there is always a radical-integer
expression for any alg-int that has some radical expression.  The proof
is in the math-fun mailing list archive, presumably in the message
that Weisstein references.
It certainly doesn't count as an academic publication, and hasn't been
reviewed by anyone not involved in the original discussion.  I can't
claim sole authorship, since others helped in the proof-creation
process.  I don't know what your standards for WP inclusion are: if
you require reviewed journal articles, you get higher accuracy; but you
lose a lot of informal math/CS that only appears on the net.
I did have one inquiry from someone a few years ago, and I was able to
dredge up the article.  But I can't find it after a short search, and my
memory is that the article wasn't publication suitable: it assumed too
much of the prior background discussion on the mailing list, and my
proof style is very informal.
I can make a search for the article if you want to try rewriting it
for general consumption, but I don't have time for a rewrite now.
I am slightly miffed at being described as "not a number theorist",
since I did a lot of factoring work in the 1970s.  But I make no
claims of error-free-ness: If you need vetting of the math, you'll
have to find your own reviewer.
Rich Schroeppel
*I'm sorry about the "not a number theorist" remark. But if you look, you'll see what I said is you didn't seem to be a number theorist, in the sense that your name had come up in discussion as a cryptographer rather than a number theorist. It certainly wasn't any judgment on the body of your work, which I'm not qualified to comment on. --Trovatore 16:52, 12 April 2006 (UTC)
thread_title: Reply from Schroeppel