Using Newton's method with the derivative I gave above, starting at '2.56', with 125-digit accuracy, I get these results:
Code:
Iteration 1:
2.56563060960481762248282987937393497549542054464
8286747384009513257202513484139358776541568224439
0858632748703058429949739842
Iteration 2:
2.56652456406646578056167108050564457088812861758
3395279837174966833713944002734979909689831246012
9867563762003639542486276154
Iteration 3:
2.56654382347877558788567847718727886298307419078
8419782048165079419952931936539487288874036810064
7478513751291809498728151677
Iteration 4:
2.56654383217242373414113906879383451033230442147
2089648153518652897650115821467896292478300260626
1634945600572639125959828278
Iteration 5:
2.56654383217242550447509230222797944953210133882
5047777927047735001396066137472047172538770141468
7658631479727115696034317965
Iteration 6:
2.56654383217242550447509230230139048641563711587
2170285653270636448636297384205893821843247782478
0024327177286692841095113007
Iteration 7:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178603161857601313466
Iteration 8:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389675
Iteration 9:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389648
Iteration 10:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389673
Iteration 11:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389646
Iteration 12:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389671
Iteration 13:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389644
Iteration 14:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389616
Iteration 15:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389641
Iteration 16:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389613
Iteration 17:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389638
Iteration 18:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389610
Iteration 19:
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
1759025178606894344698389635
I would trust this to be good to at least 100 digit accuracy making a reliable zero of the given equation,
=
2.56654383217242550447509230230139048641563711587
2170285779503861302856886575848727073486249578262
175...
Plugging this into your equation,
=
-0.00000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000
03484126325781802162370861802...
The first 20 results generated with the above are [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 39, 86, 107, 116, 120,
1010, 3530, 4609, 5137, 5252] which has diverged from the primes by the 11th iteration.
Sorry Don, I was really hoping this would work. I’ll post the code I used below in case the computer teachers at your school or anyone else you had working on it want to check for errors or whatnot. I should also mention that finding the root at this relatively low accuracy too over an hour. Finding a few hundred digits of the zero would no doubt take days.
To find the root:
Code:
use Math::BigFloat;
$n = 1;
my $x = Math::BigFloat->new('2.56' ,125);
my $pi = Math::BigFloat->bpi(125);
while ($n < 20) {
$fx = ((sin($x**(1/2)))**(-1)-1)**(-1) / (($pi)**2 + (log(log(2*((2* ($x**(-1) +1))**(-1) +1))))**(-1))-1;
$fprimex = -(((-3*$pi**2*cos(sqrt($x))*$x**2-5*$pi**2*cos(sqrt($x))*$x-2*$pi**2*cos(sqrt($x)))*log(3*$x+2)+(3*$pi**2*cos(sqrt($x))*$x**2+5*$pi**2*cos(sqrt($x))*$x+2*$pi**2*cos(sqrt($x)))*log($x+1))*log(log(3*$x+2)-log($x+1))**2+((-3*cos(sqrt($x))*$x**2-5*cos(sqrt($x))*$x-2*cos(sqrt($x)))*log(3*$x+2)+(3*cos(sqrt($x))*$x**2+5*cos(sqrt($x))*$x+2*cos(sqrt($x)))*log($x+1))*log(log(3*$x+2)-log($x+1))+(2*sin(sqrt($x))**2-2*sin(sqrt($x)))*sqrt($x))/(sqrt($x)*((((6*$pi**4*sin(sqrt($x))**2-12*$pi**4*sin(sqrt($x))+6*$pi**4)*$x**2+(10*$pi**4*sin(sqrt($x))**2-20*$pi**4*sin(sqrt($x))+10*$pi**4)*$x+4*$pi**4*sin(sqrt($x))**2-8*$pi**4*sin(sqrt($x))+4*$pi**4)*log(3*$x+2)+((-6*$pi**4*sin(sqrt($x))**2+12*$pi**4*sin(sqrt($x))-6*$pi**4)*$x**2+(-10*$pi**4*sin(sqrt($x))**2+20*$pi**4*sin(sqrt($x))-10*$pi**4)*$x-4*$pi**4*sin(sqrt($x))**2+8*$pi**4*sin(sqrt($x))-4*$pi**4)*log($x+1))*log(log(3*$x+2)-log($x+1))**2+(((12*$pi**2*sin(sqrt($x))**2-24*$pi**2*sin(sqrt($x))+12*$pi**2)*$x**2+(20*$pi**2*sin(sqrt($x))**2-40*$pi**2*sin(sqrt($x))+20*$pi**2)*$x+8*$pi**2*sin(sqrt($x))**2-16*$pi**2*sin(sqrt($x))+8*$pi**2)*log(3*$x+2)+((-12*$pi**2*sin(sqrt($x))**2+24*$pi**2*sin(sqrt($x))-12*$pi**2)*$x**2+(-20*$pi**2*sin(sqrt($x))**2+40*$pi**2*sin(sqrt($x))-20*$pi**2)*$x-8*$pi**2*sin(sqrt($x))**2+16*$pi**2*sin(sqrt($x))-8*$pi**2)*log($x+1))*log(log(3*$x+2)-log($x+1))+((6*sin(sqrt($x))**2-12*sin(sqrt($x))+6)*$x**2+(10*sin(sqrt($x))**2-20*sin(sqrt($x))+10)*$x+4*sin(sqrt($x))**2-8*sin(sqrt($x))+4)*log(3*$x+2)+((-6*sin(sqrt($x))**2+12*sin(sqrt($x))-6)*$x**2+(-10*sin(sqrt($x))**2+20*sin(sqrt($x))-10)*$x-4*sin(sqrt($x))**2+8*sin(sqrt($x))-4)*log($x+1)));
$x = $x - ($fx / $fprimex);
print "Iteration $n: $x \n \n";
$n += 1;
}
To check to root:
Code:
use Math::BigFloat;
$n = 1;
my $x = Math::BigFloat->new('2.566543832172425504475092302301390486415637115872170285779503861302856886575848727073486249578262175' ,125);
my $pi = Math::BigFloat->bpi(125);
$fx = ((sin($x**(1/2)))**(-1)-1)**(-1) / (($pi)**2 + (log(log(2*((2* ($x**(-1) +1))**(-1) +1))))**(-1))-1;
print $fx;
To generate the primes:
Code:
use Math::BigFloat;
$n = 1;
my $x = Math::BigFloat->new('2.566543832172425504475092302301390486415637115872170285779503861302856886575848727073486249578262175' ,125);
while ($n < 21) {
$n += 1;
print int($x)," $x \n \n ";
$x = (($x/int($x))-1)**(-1);
}
Output from generating primes:
Code:
2
2.566543832172425504475092302301390486415637...
3
3.530176989714906033364148596554659995910153...
5
5.658487746918629412659711455595004495617668...
7
7.593155716863869815405210254129724962568830...
11
11.80128556631025640539791517929889399141064...
13
13.72793977888878975598933174466413738362946...
17
17.8586201455355024691937312851814766372825...
19
19.7992093341782407323601518013036272125288...
23
23.7734961135508893367847814910284223028877...
29
29.7351203154905049796724099896587367504668...
39
39.4493246736759135854493344509092908313751...
86
86.7969249962215611957434718571442165786342...
107
107.914798014555272292500524333031172756101...
116
116.965710788099919975054128977049080718021...
120
120.118778240258961127708634461338116244126...
1010
1010.28605692738992756394968843990969720229...
3530
3530.76574378238047229643409124426431277717...
4609
4609.89704549251159936514080185464580543652...
5137
5137.97799384227184795092084557134786124355...
5252
5252.58930881089776777042007289489303113638...
