
Re: Prime mystery in Euler's polynomial P(k) := k^2 k + 41
Posted:
Oct 2, 2017 1:12 AM


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 primegenerating polynomial: >> >> up to a simple change of variable (unitshift). >> >> >> almost, i.e. >> >> gg(Q1) = 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 >> coprimeness 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/121207.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*(1kronecker(163,p)/(p1))));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: > > "HardyLittlewood conjecture F", > > < http://www.gutenberg.us/articles/hardylittlewood_conjecture_f > > > The imprint isn't great: badlooking 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

