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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: 3,887
Registered: 12/13/04
Re: Prime mystery in Euler's polynomial P(k) := k^2 -k + 41
Posted: Oct 2, 2017 1:12 AM
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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 .

David Bernier




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