
Re: Prime mystery in Euler's polynomial P(k) := k^2 k + 41
Posted:
Oct 4, 2017 9:50 AM


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 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 .
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/200372241/S0025571802014187/
David Bernier

