# Seriously disagreement

Artur grafix at csl.pl
Wed Sep 3 11:09:08 CEST 2008

```Dear Seqfans,
Another sample from these same author
http://www.devalco.de/liste_fact_x^3+x^2+x+1.htm
which states that all Pythagorean primes A002144 belonging to polynomial
x^3+x^2+x+1
problem is that polynomial x^3+x^2+x+1 is factorizable (x^2+1)(x+1) and
from these reason don't contained primes.
best wishes
Artur

Artur pisze:
> P.S.
> If we will be count divisors of x (not 2x^2-1)
> we will receiving 11, 171, 2126, 24300, 266400
> Mathematica codes:
> l = 0; p = 2; a = {}; Do[k = p x^2 - 1; m = Divisors[x];
> Do[If[PrimeQ[m[[y]]], l = l + 1], {y, 1, Length[m]}];
> If[N[Log[x]/Log[10]] == Round[N[Log[x]/Log[10]]], Print[l];
>  AppendTo[a, l]], {x, 1, 100000}]; a
>
> which is A064182
>
> if we will count only prime divisors of x when 2x^2-1 is prime we will
> receiving
> 9, 75, 647, 5397, 46555
> Mathematcia codes
> l = 0; p = 2; a = {}; Do[k = p x^2 - 1;
> If[PrimeQ[k], m = Divisors[x];
>  Do[If[PrimeQ[m[[y]]], l = l + 1], {y, 1, Length[m]}]];
> If[N[Log[x]/Log[10]] == Round[N[Log[x]/Log[10]]], Print[l];
>  AppendTo[a, l]], {x, 1, 100000}]; a
>
> this last is new for ONEIS
>
> Anyone sequence  isn't these from www page
> Best wishes
> Artur
>
>
> Artur pisze:
>> Dear Seqfans,
>> I was write mathematica procedure follow Franklin sugestion that
>> numbers aren't number of primes but number of prime divisors for
>> numbers of the form 2x^2-1 and x<=10^n.
>> My result is fllowing
>> *10, 154, 1904, 21741, 238392*
>> Mathematica codes:
>> l = 0; p = 2; a = {}; Do[k = p x^2 - 1; m = Divisors[k];
>> Do[If[PrimeQ[m[[y]]], l = l + 1], {y, 1,
>> Length[m]}];If[N[Log[x]/Log[10]] == Round[N[Log[x]/Log[10]]],
>> Print[l];  AppendTo[a, l]], {x, 1, 100000}]; a
>>
>> Mayby on mentioned www is nothing wrong (as Franklin belive) but what
>> author was mean are difficult puzzle to deknotting.
>>
>> Best wishes
>> ARTUR
>>
>>
>>
>>
>>
>> franktaw at netscape.net pisze:
>>> -----Original Message-----
>>> From: Peter Pein <petsie at dordos.net>
>>>
>>>> The page starts (after a tble of contents) wit a table of x (propably
>>> upper
>>>> bound of x) in the left column and the right column has got the title
>>> "Primes".
>>>
>>> Yes, but the title of the page is "Sieving for Primes ...", not
>>> "Counting
>>> Primes ...".  In fact, the column is a count of primes - as I
>>> stated, the number
>>> of primes dividing 2x^2-1 for any x <= 10^n.  And 1 is not being
>>> counted as
>>> a prime here.
>>>
>>>> Near the bottom (numbered "4.") the first entry says that 1 is prime.
>>>
>>> This is just sloppiness.  The program is outputting 1 when no new
>>> primes are
>>> found, and the author has simply copied this to the web page.
>>>
>>>> These are unmisunterstandable (is there such an word in english
>>> language?)
>>>> statements which are wrong. There is enough space to write "prime
>>> divisors" if
>>>> one wants. But the author wrote "Primes". Therefore it is nonsense.
>>>
>>>> Sorry for my ignorance but I do not want to have to _guess_ or
>>> _search_for_
>>>> the meaning of words  when reading websites concerning mathematics.
>>>
>>> It nowhere states that the numbers are the numbers of primes for x
>>> <= 10^n.
>>> It implies that these are numbers of primes in some way associated with
>>> 2x^2-1 for x <= 10^n.  It would be (much) better if there was some
>>> explanation for exactly what is being counted; but what is there is
>>> not wrong.
>>>
>>> Showing "1" in section 4 instead of blank, or perhaps the word
>>> "none", is
>>> wrong -- but doesn't mean that the author thinks 1 is prime.
>>>
>>> I agree that the page is far from ideal, but to simply dismiss it as
>>> "nonsense" is
>>> short-sighted.  There is something of value here.  The effort
>>> required to figure
>>> it out is much less than what is required to understand a typical
>>> mathematical
>>> paper.  And I see much worse in this mailing list on a regular basis.
>>>
>>> (And no, there is no such word as "unmisunderstandable".  Say "not
>>>
>>>> Peter
>>>
>>> franktaw at netscape.net schrieb:
>>>> A closer look at this web page shows that this is counting the number
>>> of
>>>> distinct prime divisors of numbers of the form 2x^2-1 for x <= 10^n,
>>> not
>>>> the number of primes.
>>>>
>>>> Note that there can be at most one prime divisor of 2x^2-1 that does
>>>> not divide 2y^2-1 for some y < x.  Every prime divisor p except
>>> possibly
>>>> one must be < 2x (in fact, p < sqrt(2) x), at which point p divides
>>>> 2 |x-p|^2 - 1.
>>>>
>>>>
>>>> -----Original Message-----
>>>> From: Artur <grafix at csl.pl>
>>>>
>>>> Dear Seqfans,
>>>>
>>>> On www page
>>>> we can read that number of primes of the form 2x^2-1 for x equal or
>>> less
>>>> than 10^n is
>>>>
>>>> 8, 84, 815, 7922, 77250, 759077, 7492588, 74198995, 736401956,
>>>> 7319543971, 72834161467
>>>>
>>>> ...
>>>>
>>>
>>>
>>> __________ Information from ESET NOD32 Antivirus, version of virus
>>> signature database 3408 (20080902) __________
>>>
>>> The message was checked by ESET NOD32 Antivirus.
>>>
>>> http://www.eset.com
>>>
>>>
>>>
>>
>> __________ Information from ESET NOD32 Antivirus, version of virus
>> signature database 3408 (20080902) __________
>>
>> The message was checked by ESET NOD32 Antivirus.
>>
>> http://www.eset.com
>>
>>
>>
>
> __________ Information from ESET NOD32 Antivirus, version of virus
> signature database 3408 (20080902) __________
>
> The message was checked by ESET NOD32 Antivirus.
>
> http://www.eset.com
>
>
>

aj> From seqfan-owner at ext.jussieu.fr  Wed Sep  3 11:10:40 2008
aj> Date: Wed, 03 Sep 2008 11:09:08 +0200
aj> From: Artur <grafix at csl.pl>
aj> CC: franktaw at netscape.net, petsie at dordos.net, seqfan at ext.jussieu.fr
aj> Subject: Re: Seriously disagreement
aj> ...
aj> Another sample from these same author
aj> http://www.devalco.de/liste_fact_x^3+x^2+x+1.htm
aj> which states that all Pythagorean primes A002144 belonging to polynomial
aj> x^3+x^2+x+1
aj> problem is that polynomial x^3+x^2+x+1 is factorizable (x^2+1)(x+1) and
aj> from these reason don't contained primes.

The statement in the web page is much fuzzier than that. It states that
"All primes of the form 4k+1 are *on* the polynom (x+1)(x^2+1)", roughly
to be interpreted as that the polynom is a generator of all of these. According
to the program immediatly below that statment, the claim is
"All primes of A002144 are divisors of integers of the cubic form (x+1)(x^2+1)"

As you observed that the polynomial factorizes, there is not much substance
left, because the claim then factorizes into "All primes of A002144 are divisors
of either x+1 or x^2+1," where of course (what else) all primes are anyway of the
form x+1.

Richard

aj> From seqfan-owner at ext.jussieu.fr  Tue Sep  2 21:51:22 2008
aj> Date: Tue, 02 Sep 2008 20:58:47 +0200
aj> From: Artur <grafix at csl.pl>
aj> To: seqfan <seqfan at ext.jussieu.fr>
aj> Subject: Seriously disagreement
aj>
aj> ...
aj> On www page
aj> we can read that number of primes of the form 2x^2-1 for x equal or less
aj> than 10^n is
aj> 8, 84, 815, 7922, 77250, 759077, 7492588, 74198995, 736401956,
aj> 7319543971, 72834161467
aj> my result by Mathematica is
aj> 7,45,303,2202,17185,141444,1200975
aj> disagreement of first position we can eliminated if we agree that 1 is
aj> prime but what with rest ???
aj> ...

1, 7, 17, 31, 49, 71, 97, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 647, 721, 799...

The list of all the primes of a(3)= 815 is shown with the program in
http://www.devalco.de/2x%5E2-1.htm .

For a(1)=8 consider the example of 8 primes
7, 17, 31, 71, 97, 127, 23, 199 at the start of the web page, which have been generated
from the sublist A056220= 1, 7, 17, 31, 49, 71, 97, 127, 161, 199 with numbers <=199
The algorithm scans the A056220 from the left to the right, and maintains a
"product of all primes found so far" in a variable z, initializes with 1. Moving on
to the right in the list, the new "candidate" 2x^2-1 is reduced via the gcd with z
to either full factorization (no new prime found, z unchanged) or to a new prime
which is added to the results. Start of the sieve: consider x=2, 2x^2-1=7 reduced w.r.t. z=1
which puts 7 into z and 7 into the output. Move to the right, x=3, 17, which is
co-prime to 7, so the new z is 7*17 and 17 is an output. Move to the right, x=4,
2x^2-1=31, which is is co-prime to 7*17, so the new z is 7*17*31, 31 is output.
Move to the right, x=5, 2x^1-1=49, which is 0 modulo the 7 in z, no output and z kept
at 7*17*31. Move to the right, x=6, 2x^2-1=71, which is prime, which generates z=7*17*31*71,
and puts 71 in the output. At x=7, 97 is again prime and output, z=7*17*31*71*97.

The next "interesting" point is at x=9, 2x^2-1=161, which has gcd(7*17*31*71*97,161)=7,
At that point the algorithm deviates from just reproducing A066436.

I am not sure whether a non-algorithmic description of the form "Primes of the form
2x^2-1 or primes which are divisors of 2x^2-1, x<=10^n such that ..." is possible.

Some trivial bounds on the a(n) thus generated are given by
(1) less than total count of x with 1<x<=10^n, in the original list.
(2) less than total number of primes in 1<p<=2*(10^n)^2-1, see A006680, that is
(3) larger than or equal to the count of primes in A066436 up to 2*(10^n)^2-1.

Richard

```