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Development of
Algorithmic Constructions |
19:55:19
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Prime sieving for the polynomial f(n)=4n2+1
0. Abstract
1. Mathematical theory
2. Description of the basic algorithm
3. Programming and algorithms
a) Programing of the basic algorithm
b) little improvement of the basic algorithm
c) Improved programming for speed, Hensel-Lifting
d) Used programm for the investigation in C++ with Heap
4. Results of the distribution of the primes
a) Table up to 1012 and table up to 240
b) Graphic of the distribution of the primes
c) Graphic of the proportion between the "reducible" and the "big" primes
d) Table of the distribution of the primes mod 6 and mod 8
5. Runtime of the algorithm
a) Table of the amount of divisions
b) Estimation of the runtime
6. Sequences of primes and reference to Njas-Sequences
a) Primes of form 4n2 + 1
b) unsorted list of Primes and 1
c) Primitive prime factors of the sequence k2 + 1 in the order that they are found
d) Prime factors of numbers of the form x2 + 1 which themselves are not of this form
e) Primes congruent to 1 or 2 modulo 4; or, primes of form x2+y2; or, -1 is a square mod p
f) number of primes of the form x2 + 1 < 10n
g) number of distinct prime divisors (taken together) of numbers of the form x2+1 for x<=10n
h) Pythagorean primes: primes of form 4n + 1
i) Numbers n such that n2 + 1 is prime
7. Links
a) A Sieve Method for Factoring Numbers of the Form n2 + 1, by Daniel Shanks, 1959
b) On the Conjecture of Hardy and Littlewood concerning the Number of Primes of the form n2+a, by Daniel Shanks, 1960
c) On the Gaussian Primes on the Line Im(x)=1, by M. C. Wunderlich, 1973
d) On Primes Represented by Quadratic Polynomials, by Stephan Baier and Liangyi Zhao, 2007
e) Search for primes of the form n2 + 1 by Marek Wolf, 2010
f) Batemann Horn Conjecture
g) Gaußsche Zahl
h) Gaussian Integer from MathWorld
i) Gaussian Prime from MathWorld
0. Abstract:
The investigation for the distribution of the primes concerning the polynomial f(n)=4n2+1
is very closed combined with the polynomial f(m)=m2+1 because the first polynomial is the result
of a substitution by m=2n for f(m).
Nevertheless the order of the primes is different and the construction of the algorithm is slightly a little
bit different too. The prime 2 could be negligated and the appearance of the second position of the primes
could not be used.
Calculation of the distribution of the primes was done for n up to 240 2014 on Athlon 64 600e processor on one core with 16 Gbyte Ram and 2 Terabyte of disk.
The program needed nearly a month for the calculation.
Only the distribution of the primes was logged and the calculated primes were not safed.
1. Mathematical theory
Let f(x) = 4x2 + 1 with x element of N
The following lemmas explain the mathematical background which is used for the described algorithms.
a) Lemma: If p | f(x) then also p | f(x+p)
p | f(x) <=> 4x2 + 1 = 0 mod p
p | f(x+p) <=> 4(x+p)2 + 1 = 0 mod p
<=> 4(x2 + 2xp + p2) + 1 = 0 mod p
<=> 4x2 + 1 = 0 mod p
Thus if p | f(x) then p | f(x+p)
b) Lemma: If p | f(x) then also p | f(-x)
p | f(x) <=> 4x2 + 1 = 0 mod p
p | f(-x) <=> 4(-x)2 + 1 = 0 mod p
<=> 4x2 + 1 = 0 mod p
Thus if p | f(x) then p | f(-x)
c) Lemma: If p | f(-x) then also p | f(-x+p)
This is a simple conclusion of b) and c) and means that
if p is a divisor of f(x) then p appears periodically concerning the function values of f(x)=4x2+1
at f(x) and f(-x) with the period length p
d) Lemma: f(x) mod 4 = {1}
f(0)=1, f(1)=5 = 1 mod 4, f(2)=17 = 1 mod 4, f(3)=37 = 1 mod 4
e) Lemma: If p | f(x) with p is a prime then jacobi (-1, p) = 1
because the discriminant of f(x)=4x2+1 is -1
If p divides f(x), then p divides (4 k2 + 1) which implies
that p is the sum of two squares which implies that p = 1 mod 4
which implies that jacobi(-1,p) = 1.
f) Lemma: If p is a primitiv prime factor resp. if p | f(x) with the smallest x>0
then p > x
The divisor p of f(x) appears periodically in the sequence of f(x) with x=0 up to oo
respectively if p | f(x) then p | f(x+p) and especially p | f(x-p).
supposing that p<=x then p would be found earlier by the described algorithm and would be sieved out.
Therefore if p divides f(x) then p is greater than x
g) Lemma: If p is a primitiv prime factor resp. if p | f(x) with the smallest x>0
then p > 2x with p>2
f(x)=f(-x)
if p<=2x then p has to appear as divisor in the interval [f(-x), f(x)]
especially in the interval [f(0), f(x)] because the function terms of f(-x) are the same as f(x).
Lemma f) shows that this is not possible.
h) Lemma: Let f*(y) the natural number which remains
when f(y) is divided as often as possible by f*(x) with x from 0 up to y-1
if f*(x)>1 then f*(x) is a primitiv prime factor and a Stormer prime
Supposing f*(y) is not a prime, f*(y)=p*q with p, q element N greater than 1
p > 2y, q > 2y (Lemma g))
f(y)=y2+1 > f*(y) = p*q > (2y)(2y) = 4y2 which is a contradiction.
i) Lemma: If f*(y) is the natural number which remains when f(y) is divided as often as possible
by the second appearance of the prime f*(x) with x from 0 up to y-1
and f*(y) > 1 then f*(y) is a prime
This proof is missing.
j) The Hensel-lifting explains for every p|f(x) where p2|f(y) p3|f(z) ... pn|f(n) can be found.
k) Lemma: if p is a prime without the 2 and with p | f(x)=x2+1 then p | f(y)=4y2+1.
The substitution with x=2y gives this result.
Therefore all primes without the 2 which are divisor of the f(x)=x2+1 are a divisor of f(y)=4y2+1.
Nevertheless the sequence of the primes found by the described algorithm is not the same and
the Lemma i) is not true for the function f(y)=4y2+1
l) With the help of the chinese remainder it is possible to calculate a x
where f(x) is either a prime or the product of new primes
and f(x) is not divisible by the primes found before.
Example:
The first prime values for the polynom fx)=x2+1 are
f(0)=1
f(1)=2
f(2)=5
f(3)=1
f(4)=17
Supposing that the sequence f(5)-f(oo)=1
p1=2
p2=5
p3=17
are the first primes
choose a special x1, x2 and x3 for the primes
x1=2
x2=4
x3=5
(Other values are also possible, f(xi) should not be divisible by pi)
With the help of the chinese remainder theorem you could solve the equitations :
x=0 mod 2 f(0)=1 not divisible by 2
x=4 mod 5 f(4)=17 not divisible by 5
x=5 mod 17 f(5)=26 not divisible by 17
The solution is x=124 mod 170
f(124) is not divisible by the primes 2, 5, 17
f(124)=15377 which is by chance a primes.
By this way you could calculate a special x taking the first primes of the sequence calculated by the algorithm
and you get as result a special f(x) which is either a prime or consists of primes which are not yet in the sequence.
m) Lemma: if x+I is divisible by a+bI then x-I is divisible by a-bI and vice visa.
Proof: (x+I) / (a+bI) = (x+I)(a-bI)/(a2+b2) = (ax+b+aI-bxI)/(a2+b2) = (ax+b)/(a2+b2) + (a-bx)I / (a2+b2) = c + dI with x, a, b, c, d element of Z
(x-I) / (a-bI) = (x-I)(a+bI)/(a2+b2) = (ax+b-aI+bxI)/(a2+b2) = (ax+b)/(a2+b2) - (a-bx)I / (a2+b2) = c - dI
if (a+bI) / (c+dI) = e+fI then
(a-bI) / (c-dI) = e-fI
the proof is analogue to the sentence before.
n) Lemma: The sequence of primes with p = a2 + b2 is infinite.
p(x)= x2+1 = (x+I)(x-I)
Proof: Let p*(x) the prime > 2 for x > 1 which remains
when p(x) is divided as often as possible by f*(x) with x from 0 up to x-1
Then p*(x) could be written according Lemma q) as p*(x)=(a+bI)(a-bI) = a2+b2 with a, b element of Z
As the sequence of reducible primes is infinite (Lemma l)
the sequence of these primes as sum of two quadratic numbers is also infinite.
2. Description of the basic algorithm
a) Create a list from x=1 to x=list_max=100 with f(x)=4x2+1
f(1)= 5
f(2)= 17
f(3)= 37
f(4)= 65 = 5*13
f(5)= 101
f(6)= 145 = 5*29
f(7)= 197
f(8)= 257
f(9)= 325 = 52*13
f(10)= 401
f(11)= 485 = 5*97
f(12)= 577
f(13)= 677
f(14)= 785 = 5*157
f(15)= 901 = 17*53
f(16)= 1025 = 52*41
f(17)= 1157 = 13*89
f(18)= 1297
f(19)= 1445 = 5*172
f(20)= 1601
f(21)= 1765 = 5*353
f(22)= 1937 = 13*149
f(23)= 2117 = 29*73
f(24)= 2305 = 5*461
f(25)= 2501 = 41*61
f(26)= 2705 = 5*541
f(27)= 2917
f(28)= 3137
f(29)= 3365 = 5*673
f(30)= 3601 = 13*277
f(31)= 3845 = 5*769
f(32)= 4097 = 17*241
f(33)= 4357
f(34)= 4625 = 53*37
f(35)= 4901 = 132*29
f(36)= 5185 = 5*17*61
f(37)= 5477
f(38)= 5777 = 53*109
f(39)= 6085 = 5*1217
f(40)= 6401 = 37*173
f(41)= 6725 = 52*269
f(42)= 7057
f(43)= 7397 = 13*569
f(44)= 7745 = 5*1549
f(45)= 8101
f(46)= 8465 = 5*1693
f(47)= 8837
f(48)= 9217 = 13*709
f(49)= 9605 = 5*17*113
f(50)= 10001 = 73*137
f(51)= 10405 = 5*2081
f(52)= 10817 = 29*373
f(53)= 11237 = 17*661
f(54)= 11665 = 5*2333
f(55)= 12101
f(56)= 12545 = 5*13*193
f(57)= 12997 = 41*317
f(58)= 13457
f(59)= 13925 = 52*557
f(60)= 14401
f(61)= 14885 = 5*13*229
f(62)= 15377
f(63)= 15877
f(64)= 16385 = 5*29*113
f(65)= 16901
f(66)= 17425 = 52*17*41
f(67)= 17957
f(68)= 18497 = 53*349
f(69)= 19045 = 5*13*293
f(70)= 19601 = 17*1153
f(71)= 20165 = 5*37*109
f(72)= 20737 = 89*233
f(73)= 21317
f(74)= 21905 = 5*13*337
f(75)= 22501
f(76)= 23105 = 5*4621
f(77)= 23717 = 37*641
f(78)= 24337
f(79)= 24965 = 5*4993
f(80)= 25601
f(81)= 26245 = 5*29*181
f(82)= 26897 = 13*2069
f(83)= 27557 = 17*1621
f(84)= 28225 = 52*1129
f(85)= 28901
f(86)= 29585 = 5*61*97
f(87)= 30277 = 13*17*137
f(88)= 30977
f(89)= 31685 = 5*6337
f(90)= 32401
f(91)= 33125 = 54*53
f(92)= 33857
f(93)= 34597 = 29*1193
f(94)= 35345 = 5*7069
f(95)= 36101 = 13*2777
f(96)= 36865 = 5*73*101
f(97)= 37637 = 61*617
f(98)= 38417 = 41*937
f(99)= 39205 = 5*7841
f(100)= 40001 = 13*17*181
b) f(1)=5
Divide f(1+k*5) / 5 with 1+k*5<=liste_max, k element N
Dvide f(-1+k*5) / 5 as often as there is no factor 5 in the result.
c) f(2)=17
Divide f(2+k*17) / 17 as often as there is no factor 17 in the result.
Divide f(-2+k*17) / 17 as often as there is no factor 17 in the result.
d) Go for x from 3 to liste_max
if f´(x) > 1
Divide f(x+k*f´(x)) / f´(x) and
Divide f(-x+k*f´(x)) / f´(x)
as often as there is no factor f´(x) in the result.
f´(x) is the prime which remains after dividing by f(1), f´(2) up to f´(x-1)
You get an unsorted list of primes.
3. a) Programming of the basic algorithm
- liste_max:=1000;
siebung:=proc (stelle, p)
begin
while (stelle<=liste_max) do
erg:=liste[stelle];
repeat
erg:=erg /p;
until (erg mod p>0) end_repeat;
liste[stelle]:=erg;
stelle:=stelle+p;
end_while;
end_proc;
for x from 1 to liste_max do
liste [x]:=4*x2+1;
end_for;
for x from 1 to liste_max do
p:=liste[x];
print (x, p);
if (p>1) then
siebung (x, p);
siebung (p-x, p);
end_if;
end_for;
3. b) little improvement of the basic algorithm
- liste_max:=1000;
anz:=0;
siebung:=proc (stelle, p)
begin
while (stelle<=liste_max) do
erg:=liste[stelle];
repeat
erg:=erg /p;
until (erg mod p>0) end_repeat;
liste[stelle]:=erg;
stelle:=stelle+p;
end_while;
end_proc;
for x from 1 to liste_max do
liste [x]:=4*x2+1;
end_for;
for x from 1 to liste_max/3 do
p:=liste[x];
if (p>1) then
anz:=anz+1;
siebung (x+p, p);
siebung (p-x, p);
end_if;
end_for;
for x from round (liste_max/3)+1 to liste_max do
p:=liste[x];
if (p>1) then
anz:=anz+1;
siebung (p-x, p);
end_if;
end_for;
print (anz);
3. c) Improved programming for speed, Hensel-Lifting
- liste_max:=1000;
anz:=0;
f:=proc (x)
begin
return (4*x2+1);
end;
fd:=proc (x)
begin
return (8*x);
end;
siebung:=proc (s, p, a)
begin
while (s<=liste_max) do
liste[s]:=liste[s]/p;
s:=s+a;
end_while;
end_proc;
hensel:=proc (s, p)
begin
a:=p;
siebung (s, p, a);
inv:=fd(s)^(-1) mod p;
repeat
a:=a*p;
s:=s-f(s)*inv;
while (s<0) do
s:=s+a;
end_while;
siebung (s, p, a);
until s>liste_max end_repeat;
end_proc;
for x from 1 to liste_max do
liste [x]:=f (x);
end_for;
for x from 1 to liste_max/3 do
p:=liste[x];
// print (x, p);
if (p>1) then
anz:=anz+1;
s:=x+p;
if (s<=liste_max) then
hensel (s, p);
end_if;
s:=p-x;
if (s<=liste_max) then
hensel (s, p);
end_if;
end_if;
end_for;
for x from round (liste_max/3)+1 to liste_max do
p:=liste[x];
// print (x, p);
if (p>1) then
anz:=anz+1;
s:=p-x;
if (s<=liste_max) then
hensel (s, p);
end_if;
end_if;
end_for;
print ("Anzahl = ", anz);
4. a) Results of the distribution of the primes
Legend of the tables:
Column A = exponent concerning base 10 resp. base 2
Column B = is the interval [0,x]
Column C = all primes resp. D+E, the amount of the primes depends of p(x) for the interval [0,x] and is not calculated by primes < x
Column D = primes of the form p=4x2+1
Column E = reducible primes with p | 4x2+1 and p < 4x2+1, counted by the first appearance resp. for the smallest x.
The numbers in the rows F-K are rounded with 6 digits after the decimal point.
A | B | C | D | E | F | G | H | I | J | K | | 10^n | x | all Primes | P(x)=4x^2+1 | P(x) | 4x^2+1 | C/B | D/B | E/B | C(n) / C(n-1) | D(n) / D(n-1) | E(n) / E(n-1) | | 1 | 10 | 9 | 7 | 2 | 0,900000 | 0,700000 | 0,200000 | | | | | 2 | 100 | 87 | 33 | 54 | 0,870000 | 0,330000 | 0,540000 | 9,666667 | 4,714286 | 27,000000 | | 3 | 1.000 | 836 | 208 | 628 | 0,836000 | 0,208000 | 0,628000 | 9,609195 | 6,303030 | 11,629630 | | 4 | 10.000 | 8.000 | 1.558 | 6.442 | 0,800000 | 0,155800 | 0,644200 | 9,569378 | 7,490385 | 10,257962 | | 5 | 100.000 | 78.124 | 12.390 | 65.734 | 0,781240 | 0,123900 | 0,657340 | 9,765500 | 7,952503 | 10,203974 | | 6 | 1.000.000 | 766.585 | 102.204 | 664.381 | 0,766585 | 0,102204 | 0,664381 | 9,812414 | 8,248910 | 10,107114 | | 7 | 10.000.000 | 7.556.731 | 872.120 | 6.684.611 | 0,755673 | 0,087212 | 0,668461 | 9,857656 | 8,533130 | 10,061412 | | 8 | 100.000.000 | 74.771.106 | 7.605.407 | 67.165.699 | 0,747711 | 0,076054 | 0,671657 | 9,894636 | 8,720597 | 10,047810 | | 9 | 1.000.000.000 | 741.554.656 | 67.420.596 | 674.134.060 | 0,741555 | 0,067421 | 0,674134 | 9,917663 | 8,864824 | 10,036880 | | 10 | 10.000.000.000 | 7.366.252.759 | 605.573.968 | 6.760.678.791 | 0,736625 | 0,060557 | 0,676068 | 9,933526 | 8,982032 | 10,028686 | | 11 | 100.000.000.000 | 73.261.462.211 | 5.496.160.249 | 67.765.301.962 | 0,732615 | 0,054962 | 0,677653 | 9,945554 | 9,075952 | 10,023446 | | 12 | 1.000.000.000.000 | 729.280.694.469 | 50.315.070.934 | 678.965.623.535 | 0,729281 | 0,050315 | 0,678966 | 9,954493 | 9,154586 | 10,019370 | | | | | | | | | | | | | | | | | | | | | | | | | | A | B | C | D | E | F | G | H | I | J | K | | 2^n | x | all Primes | P(x)=4x^2+1 | P(x) | 4x^2+1 | C/B | D/B | E/B | C(n) / C(n-1) | D(n) / D(n-1) | E(n) / E(n-1) | | 1 | 2 | 2 | 2 | 0 | 1,000000 | 1,000000 | 0,000000 | | | | | 2 | 4 | 4 | 3 | 1 | 1,000000 | 0,750000 | 0,250000 | 2,000000 | 1,500000 | | | 3 | 8 | 8 | 6 | 2 | 1,000000 | 0,750000 | 0,250000 | 2,000000 | 2,000000 | 2,000000 | | 4 | 16 | 15 | 9 | 6 | 0,937500 | 0,562500 | 0,375000 | 1,875000 | 1,500000 | 3,000000 | | 5 | 32 | 30 | 13 | 17 | 0,937500 | 0,406250 | 0,531250 | 2,000000 | 1,444444 | 2,833333 | | 6 | 64 | 58 | 23 | 35 | 0,906250 | 0,359375 | 0,546875 | 1,933333 | 1,769231 | 2,058824 | | 7 | 128 | 112 | 42 | 70 | 0,875000 | 0,328125 | 0,546875 | 1,931034 | 1,826087 | 2,000000 | | 8 | 256 | 220 | 69 | 151 | 0,859375 | 0,269531 | 0,589844 | 1,964286 | 1,642857 | 2,157143 | | 9 | 512 | 441 | 113 | 328 | 0,861328 | 0,220703 | 0,640625 | 2,004545 | 1,637681 | 2,172185 | | 10 | 1.024 | 855 | 211 | 644 | 0,834961 | 0,206055 | 0,628906 | 1,938776 | 1,867257 | 1,963415 | | 11 | 2.048 | 1.679 | 392 | 1.287 | 0,819824 | 0,191406 | 0,628418 | 1,963743 | 1,857820 | 1,998447 | | 12 | 4.096 | 3.323 | 712 | 2.611 | 0,811279 | 0,173828 | 0,637451 | 1,979154 | 1,816327 | 2,028749 | | 13 | 8.192 | 6.595 | 1.300 | 5.295 | 0,805054 | 0,158691 | 0,646362 | 1,984652 | 1,825843 | 2,027959 | | 14 | 16.384 | 13.045 | 2.458 | 10.587 | 0,796204 | 0,150024 | 0,646179 | 1,978014 | 1,890769 | 1,999433 | | 15 | 32.768 | 25.880 | 4.614 | 21.266 | 0,789795 | 0,140808 | 0,648987 | 1,983902 | 1,877136 | 2,008690 | | 16 | 65.536 | 51.435 | 8.417 | 43.018 | 0,784836 | 0,128433 | 0,656403 | 1,987442 | 1,824231 | 2,022853 | | 17 | 131.072 | 102.141 | 15.866 | 86.275 | 0,779274 | 0,121048 | 0,658226 | 1,985827 | 1,884995 | 2,005556 | | 18 | 262.144 | 203.029 | 29.842 | 173.187 | 0,774494 | 0,113838 | 0,660656 | 1,987733 | 1,880877 | 2,007383 | | 19 | 524.288 | 403.734 | 56.533 | 347.201 | 0,770061 | 0,107828 | 0,662233 | 1,988553 | 1,894411 | 2,004775 | | 20 | 1.048.576 | 803.573 | 106.786 | 696.787 | 0,766347 | 0,101839 | 0,664508 | 1,990353 | 1,888914 | 2,006869 | | 21 | 2.097.152 | 1.599.253 | 203.024 | 1.396.229 | 0,762583 | 0,096809 | 0,665774 | 1,990178 | 1,901223 | 2,003810 | | 22 | 4.194.304 | 3.184.946 | 387.307 | 2.797.639 | 0,759350 | 0,092341 | 0,667009 | 1,991521 | 1,907691 | 2,003711 | | 23 | 8.388.608 | 6.344.870 | 739.670 | 5.605.200 | 0,756367 | 0,088176 | 0,668192 | 1,992144 | 1,909777 | 2,003547 | | 24 | 16.777.216 | 12.645.628 | 1.416.634 | 11.228.994 | 0,753738 | 0,084438 | 0,669300 | 1,993048 | 1,915224 | 2,003317 | | 25 | 33.554.432 | 25.207.257 | 2.716.921 | 22.490.336 | 0,751235 | 0,080971 | 0,670264 | 1,993357 | 1,917871 | 2,002881 | | 26 | 67.108.864 | 50.260.969 | 5.218.925 | 45.042.044 | 0,748947 | 0,077768 | 0,671179 | 1,993909 | 1,920897 | 2,002729 | | 27 | 134.217.728 | 100.239.238 | 10.044.584 | 90.194.654 | 0,746841 | 0,074838 | 0,672003 | 1,994375 | 1,924646 | 2,002455 | | 28 | 268.435.456 | 199.958.544 | 19.352.154 | 180.606.390 | 0,744904 | 0,072092 | 0,672811 | 1,994813 | 1,926626 | 2,002407 | | 29 | 536.870.912 | 398.940.000 | 37.339.023 | 361.600.977 | 0,743084 | 0,069549 | 0,673534 | 1,995114 | 1,929450 | 2,002149 | | 30 | 1.073.741.824 | 796.054.419 | 72.139.394 | 723.915.025 | 0,741383 | 0,067185 | 0,674198 | 1,995424 | 1,932011 | 2,001972 | | 31 | 2.147.483.648 | 1.588.702.041 | 139.535.722 | 1.449.166.319 | 0,739797 | 0,064976 | 0,674821 | 1,995720 | 1,934251 | 2,001846 | | 32 | 4.294.967.296 | 3.171.017.355 | 270.187.319 | 2.900.830.036 | 0,738310 | 0,062908 | 0,675402 | 1,995980 | 1,936331 | 2,001723 | | 33 | 8.589.934.592 | 6.330.108.091 | 523.695.184 | 5.806.412.907 | 0,736922 | 0,060966 | 0,675955 | 1,996239 | 1,938267 | 2,001638 | | 34 | 17.179.869.184 | 12.637.730.074 | 1.016.029.275 | 11.621.700.799 | 0,735613 | 0,059141 | 0,676472 | 1,996448 | 1,940116 | 2,001528 | | 35 | 34.359.738.368 | 25.233.098.862 | 1.973.029.795 | 23.260.069.067 | 0,734380 | 0,057423 | 0,676957 | 1,996648 | 1,941903 | 2,001434 | | 36 | 68.719.476.736 | 50.386.282.403 | 3.834.641.364 | 46.551.641.039 | 0,733217 | 0,055801 | 0,677416 | 1,996833 | 1,943529 | 2,001354 | | 37 | 137.438.953.472 | 100.621.550.807 | 7.458.662.438 | 93.162.888.369 | 0,732118 | 0,054269 | 0,677849 | 1,997003 | 1,945074 | 2,001280 | | 38 | 274.877.906.944 | 200.957.079.562 | 14.518.923.630 | 186.438.155.932 | 0,731078 | 0,052820 | 0,678258 | 1,997157 | 1,946585 | 2,001206 | | 39 | 549.755.813.888 | 401.371.984.579 | 28.282.415.899 | 373.089.568.680 | 0,730091 | 0,051445 | 0,678646 | 1,997302 | 1,947969 | 2,001144 | | 40 | 1.099.511.627.776 | 801.714.900.206 | 55.130.064.460 | 746.584.835.746 | 0,729155 | 0,050141 | 0,679015 | 1,997436 | 1,949270 | 2,001087 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
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