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From: Ludovicus on 28 Jul 2010 08:34
On 27 jul, 19:50, Rick Decker <rdec...(a)hamilton.edu> wrote:
> On 7/27/10 12:21 PM, Ludovicus wrote:>
"The set of least divisors of  x^2 + x + A is the set of prime
numbers, (A = an odd integer.)
for each A  testing  x = 0,1,2,3...(Non negative integers)."
Example: 79 is the least divisor of x^2 + x + 26149427 .
(Its  first appearance as minimum divisor)
>
> Oh? What about x^2 + x + 6239 and x = 1?
> Rick

It is true that for x =1 that trinom is divisible by 79
but you must test other values: x = 0,1,2,3,4...
In that case you will find that the minimum prime divisor is 5.
Ludovicus
From: sttscitrans on 28 Jul 2010 12:52
On 28 July, 07:15, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <1a62f9ae-a1cd-4e61-956e-a1914e940...(a)l14g2000yql.googlegroups.com>,
>
>
>
>
>
>  "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net> wrote:
> > On 27 July, 17:21, Ludovicus <luir...(a)yahoo.com> wrote:
> > > "The set of least divisors of  x^2 + x + A is the set of prime
> > > numbers, (A = an odd integer.)
> > > for each A  testing  x = 0,1,2,3...(Non negative integers)."
>
> > > Example: 79 is the least divisor of x^2 + x + 26149427 .
> > >  (Its  first appearance as minimum divisor)
> > > Ludovicus
>
> > Do you mean that for every A, x^2 + x + A
> > there is a least prime divisor for x=0, 1,2, ...  and you are
> > considering the smallest of all the least prime divisors?
> > Is A = 261494427  the first A for which
> > 79 is the smallest prime divisor ?
>
> > Is A=47 the "first appearance" of 7 as the
> > smallest divisor ?
>
> x^2 + x is always even, so if A is odd then 2 doesn't divide
> x^2 + x + A.

Yes, so the conjecture that every prime is the
smallest prime divisor for some A must be false.

> x^2 + x is always 0 or 2 mod 3, so if A is 2 mod 3 then
> 3 doesn't divide x^2 + x + A.
>
> x^2 + x is 0, 1, or 2 mod 5, so if A is 1 or 2 mod 5 then
> 5 doesn't divide x^2 + x + A.
>
> Putting these together, if A is 11 or 17 mod 30 then
> 2, 3, and 5 are not divisors of x^2 + x + A.
>
> x^2 + x is 0, 2, 5, or 6 mod 7. So A must be
> 0, 1, 2, or 5 mod 7 for 7 to divide x^2 + x + A.
>
> Looking at the numbers that are 11 or 17 mod 30,
> viz., 11, 17, 41, 47, ..., the first one that is in an
> admissible class mod 7 is 47.
>
> So, yes, it would appear that A = 47 is the first appearance
> of 7 as the smallest divisor.

OK, thanks for confirmation
From: Kermit Rose on 28 Jul 2010 12:32
"The set of least divisors of x^2 + x + A is the set of prime
numbers, (A = an odd integer.)
for each A testing x = 0,1,2,3...(Non negative integers)."

Example: 79 is the least divisor of x^2 + x + 26149427 .
(Its first appearance as minimum divisor)
Ludovicus


***********

The least divisor of a number > 1, is, by nature,
a prime number.


x^2 + x + A

For x = 0, (A + 0); x^2 + x + A = {1,3,5,7,....}
For x = 1, (A + 2): x^2 + x + A = {3,5,7,...}
For x = 2, (A + 3): x^2 + x + A = {4,6,8,10,...}

For A ranging over the odd integers,

x^2 + x + A ranges over all integers.

Your conjecture is trivially true.

Kermit Rose
From: Gerry Myerson on 28 Jul 2010 20:04
In article
<0c9c2cc3-b9d0-4aa1-87d9-43954552e883(a)s9g2000yqd.googlegroups.com>,
"sttscitrans(a)tesco.net" <sttscitrans(a)tesco.net> wrote:

> On 28 July, 07:15, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
> wrote:
> > In article
> > <1a62f9ae-a1cd-4e61-956e-a1914e940...(a)l14g2000yql.googlegroups.com>,
> >
> >
> >
> >
> >
> > �"sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net> wrote:
> > > On 27 July, 17:21, Ludovicus <luir...(a)yahoo.com> wrote:
> > > > "The set of least divisors of �x^2 + x + A is the set of prime
> > > > numbers, (A = an odd integer.)
> > > > for each A �testing �x = 0,1,2,3...(Non negative integers)."
> >
> > > > Example: 79 is the least divisor of x^2 + x + 26149427 .
> > > > �(Its �first appearance as minimum divisor)
> > > > Ludovicus
> >
> > > Do you mean that for every A, x^2 + x + A
> > > there is a least prime divisor for x=0, 1,2, ... �and you are
> > > considering the smallest of all the least prime divisors?
> > > Is A = 261494427 �the first A for which
> > > 79 is the smallest prime divisor ?
> >
> > > Is A=47 the "first appearance" of 7 as the
> > > smallest divisor ?
> >
> > x^2 + x is always even, so if A is odd then 2 doesn't divide
> > x^2 + x + A.
>
> Yes, so the conjecture that every prime is the
> smallest prime divisor for some A must be false.

Only because OP insists on A being odd,
and only because OP asks for the set of prime numbers,
not the set of odd prime numbers.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Gerry Myerson on 28 Jul 2010 20:06
In article
<1997149530.19160.1280349209044.JavaMail.root(a)gallium.mathforum.org>,
Kermit Rose <kermit(a)POLARIS.NET> wrote:

> "The set of least divisors of x^2 + x + A is the set of prime
> numbers, (A = an odd integer.)
> for each A testing x = 0,1,2,3...(Non negative integers)."
>
> Example: 79 is the least divisor of x^2 + x + 26149427 .
> (Its first appearance as minimum divisor)
> Ludovicus
>
>
> ***********
>
> The least divisor of a number > 1, is, by nature,
> a prime number.
>
>
> x^2 + x + A
>
> For x = 0, (A + 0); x^2 + x + A = {1,3,5,7,....}
> For x = 1, (A + 2): x^2 + x + A = {3,5,7,...}
> For x = 2, (A + 3): x^2 + x + A = {4,6,8,10,...}
>
> For A ranging over the odd integers,
>
> x^2 + x + A ranges over all integers.
>
> Your conjecture is trivially true.

I think you are misunderstanding the conjecture,
which is easy enough to do, given how badly it was stated.
See the other posts in this thread, where I think I succeeded
in figuring out just exactly what OP meant.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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