New finite sequence

[Chris Boyd]

Some years ago I discovered a series of less than 500 positive integers
that share a certain simple number-theoretic property. The largest numbers
in the series appear to be:

730847, 911027, 1011218, 1122558, 1153547, 1191302, 1195862,
1198823, 1200023, 1215843, 1230990, 1586343, 1607627, 1875902

There is strong numerical evidence that the series ends at 1875902.

(The sequence does not occur at http://www.research.att.com/~njas/sequences/)

Is this series already known?

Should no-one work it out, I will post an explanation of the properties of
these numbers later.

--
IAMFI

[Chris Boyd]

chrisb...@netscape.net (Chris Boyd) wrote in message news:<6e793af4.02082...@posting.google.com>...

> The largest numbers
> in the series appear to be:
>
> 730847, 911027, 1011218, 1122558, 1153547, 1191302, 1195862,
> 1198823, 1200023, 1215843, 1230990, 1586343, 1607627, 1875902
>
> There is strong numerical evidence that the series ends at 1875902.

The series consists of those natural numbers N which _cannot_ be
represented in integers by the simple quadratic form,

x^2 + My^2 = N (1)

with x >= 1, y > 1, M >= 1.

I calculate that the series has 436 members <= 1875902:

1, 2, 3, 4, 6, 7, 11, 14, 15, 23, 30, 35, 38, 39, 42, 47, 62, 71,
...

730847, 911027, 1011218, 1122558, 1153547, 1191302, 1195862,
1198823, 1200023, 1215843, 1230990, 1586343, 1607627, 1875902

1875902 looks like the largest natural number N with this property
(I have checked up to 100,000,000).

If true, every sufficiently large number is expressible as x^2 + My^2
(with x >= 1, y > 1, M >= 1), or (prosaically) as the sum of a square and a
squareful number (as defined at http://mathworld.wolfram.com/Squareful.html).

Can this be proved? Any ideas?

Eqn (1) resembles the Pell-like equation

x^2 - My^

whose integer solutions x, y for certain N may be determined from continued
fraction expansions of sqrt(M).

There is a brief discussion of the more general equation

ax^2 + by^

at http://mathworld.wolfram.com/PellEquation.html, and its solution for the
special case a = 1, c prime.

--
IAMFI

[Dave Rusin]

In article <6e793af4.02082...@posting.google.com>,

Chris Boyd <chrisb...@netscape.net> wrote:
>
>The series consists of those natural numbers N which _cannot_ be
>represented in integers by the simple quadratic form,
>
> x^2 + My^2 = N (1)
>
>with x >= 1, y > 1, M >= 1.
>
>I calculate that the series has 436 members <= 1875902:
>
> 1, 2, 3, 4, 6, 7, 11, 14, 15, 23, 30, 35, 38, 39, 42, 47, 62, 71,
>...
> 730847, 911027, 1011218, 1122558, 1153547, 1191302, 1195862,
> 1198823, 1200023, 1215843, 1230990, 1586343, 1607627, 1875902
>
>
>1875902 looks like the largest natural number N with this property
>(I have checked up to 100,000,000)

Just off the top of my head I would not have expected the set to be
finite, unless the following stronger statement were true. I have no
real reason to believe it either, but wonder whether the data you
have collected in this long search support it:

There is a finite set of natural numbers M_i with the property
that for every sufficiently large integer N there exists a
solution to at least one of the Diophantine equations
x^2 + M_i y^2 = N

So, how 'bout it? What's the largest M you needed to show that
every N > 1875902 is so represented?

dave

[Gerry Myerson]

In article <6e793af4.02082...@posting.google.com>,
chrisb...@netscape.net (Chris Boyd) wrote:

=> If true, every sufficiently large number is expressible as x^2 + My^2
=> (with x >= 1, y > 1, M >= 1), or (prosaically) as the sum of a square
=> and a squareful number (as defined at
=> http://mathworld.wolfram.com/Squareful.html).

This is not the standard definition (nor the standard spelling)
of squarefull. A squarefull number is one whose every prime factor
is a repeated factor, that is, in its prime-power factorization every
exponent is at least 2.
--
Gerry Myerson (ge...@mpce.mq.edi.ai) (i -> u for email)

[David Harden]

Gerry Myerson <ge...@mpce.mq.edi.ai.i2u4email> wrote in message news:<gerry-DEAB92.12344827082002@[137.111.1.11]>...

I thought the term for that was "powerful". Maybe both terms are used
in different places?

---- David

[Chris Boyd]

ru...@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<akec9n$gq1$1...@news.math.niu.edu>...

Thanks Dave. I think I see where you're going -- something analogous to
covering congruences? Unfortunately, what you suggest will take quite some
(computer) time. I did the initial search economically by ceasing to
analyse any N as soon as the first x^2 + M y^2 representation was found.

I have made a start with N < 1875902. There certainly doesn't seem to be a
trend towards a particularly low ceiling for M_i, which would help your
conjecture. E.g., for 400395, the most extreme example found so far (of
those N < 800000 for which an M_i exists), the lowest M_i is 1019 (592^2 +
1019*7^2). There are dozens of N in that range with minimal M_i > 500.

--
IAMFI

[Gerry Myerson]

In article <a0d492b9.02082...@posting.google.com>,
rodd...@mit.edu (David Harden) wrote:

=> Gerry Myerson <ge...@mpce.mq.edi.ai.i2u4email> wrote in message
=> news:<gerry-DEAB92.12344827082002@[137.111.1.11]>...
=> > This is not the standard definition (nor the standard spelling) of
=> > squarefull. A squarefull number is one whose every prime factor is
=> > a repeated factor, that is, in its prime-power factorization every
=> > exponent is at least 2.
=>
=> I thought the term for that was "powerful". Maybe both terms are used
=> in different places?

I think standard usage takes powerful & squarefull as synonyms.