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Topic: Prime mystery in Euler's polynomial P(k) := k^2 -k + 41
Replies: 3   Last Post: Oct 4, 2017 9:50 AM

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David Bernier

Posts: 3,884
Registered: 12/13/04
Re: Prime mystery in Euler's polynomial P(k) := k^2 -k + 41
Posted: Oct 4, 2017 9:50 AM
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On 10/02/2017 01:12 AM, David Bernier wrote:
> On 10/01/2017 04:42 PM, David Bernier wrote:
>> On 10/01/2017 11:46 AM, David Bernier wrote:
>>>
>>> gg(X):= X^2+X+41
>>>
>>> gg(.) is Euler's prime-generating polynomial:
>>>
>>> up to a simple change of variable (unit-shift).
>>>
>>>
>>> almost, i.e.
>>>
>>> gg(Q-1) = Q^2 - Q + 41 , [ Euler's polynomial in Q ].
>>> which is of the same form as Euler's k^2 - k + 41
>>> from Euler's Lucky numbers:
>>>
>>> < https://en.wikipedia.org/wiki/Lucky_numbers_of_Euler > .
>>>
>>> Modulo 41, two residue classes , k == 0 (mod 41)
>>> and  k == 1 (mod 41) yield a k^2 - k + 41 == 0 (mod 41).
>>> If k > 40, and either k == 0 or k == 1 (mod 41), then
>>> 41 divides k^2 -k +1, and this last number is not
>>> a prime.
>>>
>>> There remain 39 residue classes modulo 41 which aren't
>>> forbidden from producing primes, when k > 40.
>>>
>>> For "large" swaths of consecutive integers,
>>> I tested candidates, where a candidate,
>>> in terms of Euler's P(k) = k^2 - k+1,
>>> is a k>40 with k =/= 0 , k =/= 1 (modulo 41).
>>>
>>> These candidates are not divisible by 41.
>>>
>>> If K^2 - K +1 is prime, I give it a weight
>>> of log(K^2 - K +1).
>>>
>>> Then, I look at the sum of the weights of
>>> the primes of the form: k^2 - k+1,
>>> and the number of candidates, for k
>>> in a large range of consecutive integers
>>> [ a, b].
>>>
>>> I calculate the quotient:
>>>   (sum of weights of primes)/(number of candidates),
>>> for large intervals [a, b].
>>>
>>> This quotient approaches 6.98 over ranges [a, b]
>>> that include thousands to millions of candidates
>>> that are in fact probable primes (pseudoprimes).
>>>
>>> Example:
>>>
>>> For the range
>>> [ 3,000,000,001 ... 4,000,000,000]
>>>
>>> there are:
>>> 951,219,512 candidates X such that
>>> X^2+X+41 =/= 0 (mod 41)
>>>
>>> and there are:
>>>
>>> 151,101,437 pseudoprimes (probable primes),
>>>
>>> and the weight of the probable primes is
>>> 6,640,090,792.4
>>>
>>> and weight/candidates ~= 6.98 .
>>>
>>>
>>>
>>> I looked for patterns in prime factors of
>>>   x^2 + x + 41, when x^2 + x + 41 is composite,
>>> and found no pattern. [ equivalently, poly. k^2 - k + 41 ].
>>>
>>> So I'm puzzled as to why this 6.98 ~= 7 persists,
>>> even with x (or k) into a few billions.
>>>
>>> Could it all be explained by
>>> co-primeness to the primes from 2 to 37 inclusive?

>>
>>
>> I found a seminar handout by Edray Goins ( Purdue Math Dept)
>> on quadratic polynomials in Z[x] and a Conjecture of
>> Hardy and Littlewood known as "Conjecture F" having to
>> do with the distribution of prime numbers among the
>> values taken by f(x) = a*x^2 + b*x + c, for
>> a, b, c in Z.
>>
>> He uses the notation pi_f (x), which I'm guessing
>> is
>>
>> pi_f(x)  := | { y: 1 <= y <= x, and f(y) prime }  |
>>
>> A link to his seminar handout in PDF format is the
>> following:
>>
>> < http://www.math.purdue.edu/~egoins/seminar/12-12-07.pdf > .
>>
>> Euler's quadratic appears in the form f(x) = x^2 + x + 41,
>> with disciminant Delta = -163.
>>
>> A constant C(Delta) is defined by an infinite product involving
>> primes, the Legendre symbol, and -163 (or Delta).
>>
>> Goins writes that H.C. Williams finds
>>
>> C(-163) ~= 3.3197732
>>
>> That's what I got evaluating a partial product with the PARI/gp
>> calculator, for the record:
>>
>> pp=1.0;for(X=3,10^8,if(isprime(X),p=X;pp=pp*(1-kronecker(-163,p)/(p-1))));pp
>>
>>
>>   = 3.3197732923520903506343596548256201696
>>
>>
>> If pi_f(x) is what I think it is, the Conjecture would say that
>>
>> pi_f (x) ~= C(-163) li(x)
>>
>> or pi_f(x) ~= 3.319 li(x) ,
>>
>> for f(x) := x^2 + x + 1.
>>
>> Then, according to my calculations, to get the
>> supposed constant ~= 6.98  in the computations below,
>>
>> one takes:
>>
>> (41/39)*2*C(-163) ~= 6.9800 .
>>
>> I was looking for more references.
>>
>> There's a web page with references, some encyclopedia:
>>
>> "Hardy-Littlewood conjecture F",
>>
>> < http://www.gutenberg.us/articles/hardy-littlewood_conjecture_f >
>>
>> The imprint isn't great: bad-looking math symbols, elided references;

>
> [...]
>
> A quite recent refereed publication on quadratic polynomials in Z[x]
> and the primes they generate:
>
>
> Jacobson and Williams,
> "New Quadratic Polynomials with High Densities of Prime Values",
> Math. Comp. , 72 (2003), 499?51 .


From browsing the foregoing paper:



For f(x) := x^2 + x - 4743500755044979 [Jacobson & Williams ]

we define pi_f(x) = | { y: 0<= y <= x, and f(y) is a prime} |.

So we have a count of prime values assumed by
a quadratic polynomial in some range of
consecutive integers as arguments to the
polynomial function.



If x >= 10^8 , x^2 >= 10^16 > 47,43,500,755,044,979.

So if x>= 10^8, f(x) > 1.



I find pi_f(10^8 + 10^D) - pi_f(10^8) as follows, in terms
of D:

3 303
4 2898
5 28827
6 285657
7 2844076
8 27710324
9 258,381,919
10 2,346,163,236

Point of precision:

i used "ispseudoprime" in PARI/gp to test for being a
propbable prime. Thus, the counts above are are really
counts of pseudoprime values of
x^2 + x - 4743500755044979 for x in [ 10^8, 10^8 + 10^9] ,
the D = 9 case.

the "-4743500755044979" is taken
from Williams and Jacobson,
"New quadratic polynomials with high densities of prime values",
Math. Comp., 2003 at page:
http://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/

David Bernier





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