Marvin Ray Burns

 I've been using Maple since 1997 or so.

MaplePrimes Activity


These are replies submitted by Marvin Ray Burns

@Robert Israel

 

You were correct. At first I thought you were saying the lower value  was -1. However, it was as you correctly said (or at least clearly meant) the value of the upper one minus1.

 

@Robert Israel

 

You were correct. At first I thought you were saying the lower value  was -1. However, it was as you correctly said (or at least clearly meant) the value of the upper one minus1.

 

 That is what the even-numbered partial sums were supposed to converge to. i.e. 0.1878... However, I thought the odd numbered partial sums were supposed to converge to 1 minus what the even-numbered converged to. i.e. -0.81214...

 EG. 

In Maple,

evalf(sum((cos(Pi*n)+I*sin(Pi*n))*(cosh(ln(n)/n)+sinh(ln(n)/n)), n = 1 .. 1001)) gives -.8156017982, and

evalf(sum((cos(Pi*n)+I*sin(Pi*n))*(cosh(ln(n)/n)+sinh(ln(n)/n)), n = 1 .. 3001)) gives -.8134758429.

 That is what the even-numbered partial sums were supposed to converge to. i.e. 0.1878... However, I thought the odd numbered partial sums were supposed to converge to 1 minus what the even-numbered converged to. i.e. -0.81214...

 EG. 

In Maple,

evalf(sum((cos(Pi*n)+I*sin(Pi*n))*(cosh(ln(n)/n)+sinh(ln(n)/n)), n = 1 .. 1001)) gives -.8156017982, and

evalf(sum((cos(Pi*n)+I*sin(Pi*n))*(cosh(ln(n)/n)+sinh(ln(n)/n)), n = 1 .. 3001)) gives -.8134758429.

 

Try Digits:=1000; not Digit:=1000;.

marvinrayburns.com

 

Try Digits:=1000; not Digit:=1000;.

marvinrayburns.com

 

 

Page 5:

Let x=MRB constant.

The digits of accuracy (D.O.A.) might seem embellished by 1 digit because it is computed by the reciprocal of the magnitude of error. For instance, The Champernowne Constant = 0.1234567891011121314151617181920212223243..., and its approximation constructed from the MRB constant is (1/2)*(8647+1490*x)/(31589+24300*x).  0.1234567891011121314151617181920212223243-(1/2)*(8647+1490*x)/(31589+24300*x)= 9.79861545008*10^(-19). Thus, according to this rule, there are 19 digits worth of accuracy in the approximation. However, You could argue that counting the 0, only 18 digits actually match .

Each approximation is followed by a Maple input of what is written in the pretty print, so you can use Maple to see them for yourself.

Sierpiński Constant =   (-7652+81*x)/(-9550+1431*x)   to 17 D.O.A.  (-7652+81*x)/(-9550+1431*x)

The dimer constant =   (8*(3412+5*x))/(14629+3250*x)   to 18 D.O.A.  (8*(3412+5*x))/(14629+3250*x)

Hard Hexagon Entropy Constant =   (10*(-691+61*x))/(-5033+870*x)   to 16 D.O.A.  (10*(-691+61*x))/(-5033+870*x)

Lehmer's minimal Mahler Measure Constant (i.e. 1.17628...) =  (6*(-1729+360*x))/(-8941+2484*x)    to 17 D.O.A.  (6*(-1729+360*x))/(-8941+2484*x)

The mean Cube in Tetrahedron volume =   (30*(389+15*x))/(838403+57186*x)   to 19 D.O.A  (30*(389+15*x))/(838403+57186*x)

Paris constant =  (10/9)*(-4547+286*x)/(-4567+121*x)    to 18 D.O.A.  (10/9)*(-4547+286*x)/(-4567+121*x)

 

Page 4:

Let x=MRB constant.

The digits of accuracy (D.O.A.) might seem embellished by 1 digit because it is computed by the reciprocal of the magnitude of error. For instance, The Champernowne Constant = 0.1234567891011121314151617181920212223243..., and its approximation constructed from the MRB constant is (1/2)*(8647+1490*x)/(31589+24300*x).  0.1234567891011121314151617181920212223243-(1/2)*(8647+1490*x)/(31589+24300*x)= 9.79861545008*10^(-19). Thus, according to this rule, there are 19 digits worth of accuracy in the approximation. However, You could argue that counting the 0, only 18 digits actually match .

Each approximation is followed by a Maple input of what is written in the pretty print, so you can use Maple to see them for yourself.

 

The upper bound of Landau Constant =  (1/20)*(185372+14133*x)/(17028+1477*x)    to 19 digits of accuracy. (1/20)*(185372+14133*x)/(17028+1477*x)

Log(2) 1/75721 *( ((90019/10)*Pi+24600-2100*x))   to 19 D. O. A.. 1/75721*((90019/10)*Pi+24600-2100*x)

Polygon circumscription constant =  (2*(-21925+1608*x))/(-381505+28024*x)   to 19 D.O.A.   (2*(-21925+1608*x))/(-381505+28024*x)

One-Ninth Constant =  (8/3)*(1314+601*x)/(32759+13768*x)    to 19 D.O.A   (8/3)*(1314+601*x)/(32759+13768*x)

 

Tribonacci Constant =   (10*(6658+553*x))/(36139+3325*x)  to 18 D. O. A.. (10*(6658+553*x))/(36139+3325*x)

 The offset at which two unit disks overlap by half each's area (2*(-80081+4033*x))/(-198246+10049*x)   to 19 D.O.A..   (2*(-80081+4033*x))/(-198246+10049*x)  

The offset at which two unit disks overlap by half each's area =  (1/40)*(206581+58200*x)/(6382+1855*x)  to 18 D.O.A..(1/40)*(206581+58200*x)/(6382+1855*x)

Average distance between two points chosen at random in a unit square..=  (10*(5787+301*x))/(111401+3577*x)    to 18 D.O.A.  (10*(5787+301*x))/(111401+3577*x)

Liouville's Constant =  (15451-259*x)/(138563+7756*x)   to 18 D.O.A.. (15451-259*x)/(138563+7756*x)

Gieseking's Constant =   (1/7)*(-624689+21539*x)/(-87927+3029*x)   to 19 D.O.A   (1/7)*(-624689+21539*x)/(-87927+3029*x)

The ubiquitous Constant (2*(24411+1232*x))/(57643+2821*x)    to 17 D.O.A..  (2*(24411+1232*x))/(57643+2821*x)

The rabbit Constant =   (1/60)*(63631+51200*x)/(1487+1240*x)    to 18 D.O.A..  (1/60)*(63631+51200*x)/(1487+1240*x)

Pell Constant =   (375*(59+8*x))/(37865+6464*x)   to 18 D.O.A.   (375*(59+8*x))/(37865+6464*x)

The Reuleaux tetrahedron volume =   (4771-1684*x)/(10377+932*x)   to 17 D.O.A.  (4771-1684*x)/(10377+932*x)

Copeland-Erdős Constant =  (1/3)*(92951+8000*x)/(131783+9528*x)   to 18 D.O.A.  (1/3)*(92951+8000*x)/(131783+9528*x)

Wyler's Constant =   (-593+9*x)/(-82001+5165*x)    to 19 D.O.A.  (-593+9*x)/(-82001+5165*x)

The hexagonal Madelung constant =  (8*(1141-68*x))/(1071+2336*x)    to 16 D.O.A.  (8*(1141-68*x))/(1071+2336*x)

Thue constant in base 10 =   (32*(-37833+1927*x))/(-1409291+72184*x)   to 19 D.O.A.  (32*(-37833+1927*x))/(-1409291+72184*x)

Twin Primes Constant =   (1/2)*(-188539+9258*x)/(-142788+6961*x)   to 20 D.O.A.  (1/2)*(-188539+9258*x)/(-142788+6961*x)

Cahen's Constant =   (1040*(-339+16*x))/(-548059+26416*x)   to 18 D.O.A.  (1040*(-339+16*x))/(-548059+26416*x)

Conway's Constant =   (1/10)*(57511+15930*x)/(4523+630*x)    to 18 D.O.A.  (1/10)*(57511+15930*x)/(4523+630*x)


Conway's constant =   (10*(13066+9559*x))/(95399+99055*x)   to 21 D.O.A.  (10*(13066+9559*x))/(95399+99055*x)

The prime constant in base 10 =   (9/20)*(5334+817*x)/(5779+936*x)  to 18 D.O.A.   (9/20)*(5334+817*x)/(5779+936*x)

The mean line-in-equilateral triangle length (i.e.0.3647918433...) =  (4*(3319+570*x))/(35997+8360*x)    to 18 D.O.A..  (4*(3319+570*x))/(35997+8360*x)

Champernowne Constant =  (1/2)*(8647+1490*x)/(31589+24300*x)    to 19 digits of accuracy.  (1/2)*(8647+1490*x)/(31589+24300*x)

 

 Page 3:

Let x= MRB constant.  Each approximation is followed by a maple input so you can verify these approximations.

The digits of accuracy (D.O.A.) might seem embellished by 1 digit because it is computed by the reciprocal of the magnitude of error. For instance, The Champernowne Constant = 0.1234567891011121314151617181920212223243..., and its approximation constructed from the MRB constant is (1/2)*(8647+1490*x)/(31589+24300*x).  0.1234567891011121314151617181920212223243-(1/2)*(8647+1490*x)/(31589+24300*x)= 9.79861545008*10^(-19). Thus, according to this rule, there are 19 digits worth of accuracy in the approximation. However, You could argue that counting the 0, only 18 digits actually match .

Brun's constant =  (7273/38)*Pi-11448/19+(318/19)*x   to at least all but the last known digit. (7273/38)*Pi-11448/19+(318/19)*x

The continued fraction constant =   (9/2)*(99-73*x)/(541+48*x)  to 15 digits of accuracy.  (9/2)*(99-73*x)/(541+48*x)

The Continued Fraction Constant =   (3826+1903*x)/(6167-913*x)   to 18 D.O.A.  (3826+1903*x)/(6167-913*x)

Lochs' constant =  (105/22)*(-165+8*x)/(-811+36*x)  to 16 digits of accuracy. (105/22)*(-165+8*x)/(-811+36*x)

Norton's constant = (10*(6193+200*x))/(946338+37565*x)   to 19 digits of accuracy.  (10*(6193+200*x))/(946338+37565*x)

Gauss's Constant  =  (1/3)*(-173434+4015*x)/(-69272+1635*x)  to 17 digits of accuracy.  (1/3)*(-173434+4015*x)/(-69272+1635*x)

Artin'sConstant=  (2*(449+52*x))/(2141+1664*x)  to 17 digits of accuracy. (2*(449+52*x))/(2141+1664*x)

Dottie number =  (1/3)*(-9253+198*x)/(-4469+1664*x)  to 18 digits of accuracy. (1/3)*(-9253+198*x)/(-4469+1664*x)

QRS Constant =  (2/3)*(1927+303*x)/(2165+104*x)   to 16 digits of accuracy. (2/3)*(1927+303*x)/(2165+104*x)

Decimal expansion the associated QRS constant .=   (20*(1874+225*x))/(18278+2799*x)   to 17 D.O.A.  (20*(1874+225*x))/(18278+2799*x)

Fibonacci factorial constant=   (1/210)*(19567+5556*x)/(77+16*x)  to 18 digits of accuracy.  (1/210)*(19567+5556*x)/(77+16*x)

The lemniscate constant =  (2*(13981+2709*x))/(10819+1242*x)   to 18 digits of accuracy. (2*(13981+2709*x))/(10819+1242*x)

Madelung Constant(1.747564594...) =  -(1/10)*(292017+8050*x)/(16711+455*x)  to 18 digits of accuracy.  -(1/10)*(292017+8050*x)/(16711+455*x)

 Madelung Constant(1.747564594...)  = (5/2)*(1073-572*x)/(-1664+1505*x)   to 17 D.O.A.  (5/2)*(1073-572*x)/(-1664+1505*x)

Madelung constant =   -(2*(25687+2007*x))/(29807+117*x)   to 17 D.O.A.  -(2*(25687+2007*x))/(29807+117*x)

Lochs' constant,=  (-37333+1300*x)/(-4282+185*x) to 17 digits of accuracy.  (-37333+1300*x)/(-4282+185*x)

Wallis's constant =  (14063+465*x)/(6799-230*x)  to 16 digits of accuracy.  (14063+465*x)/(6799-230*x)

The unique point at which the Gram point is equal to its index.=  (1/20)*(10767999-350452*x)/(57+8*x) to 14 digits of accuracy.  (1/20)*(10767999-350452*x)/(57+8*x)

The Golden Ratio Conjugate(1/800)*(1080437+25600*x)/(2185+53*x)  to 20 digits of accuracy.  (1/800)*(1080437+25600*x)/(2185+53*x)

Reciprocal Fibonacci Constant=   (3/86)*(-39497+936*x)/(-409+4*x)  to 17 digits of accuracy.  (3/86)*(-39497+936*x)/(-409+4*x)

Lehmer's constant =  (1/4)*(-7811+328*x)/(-3276+37*x)   to 17 digits of accuracy. (1/4)*(-7811+328*x)/(-3276+37*x)

Mills' constant =  (423322+19585*x)/(324041+15000*x)  to 18 digits of accuracy.  (423322+19585*x)/(324041+15000*x)

Mills' Constant =  (-1557033+56441*x)/(-1191961+43687*x)    to 18 D.O.A.  (-1557033+56441*x)/(-1191961+43687*x)

The real part of the omega-1 constant =   (237*(51+2*x))/(15785+702*x)  to 16 digits of accuracy. (237*(51+2*x))/(15785+702*x)

The omega2 Constant =  (1/16)*(25371-5*x)/(996+215*x)  to 18 digits of accuracy.   (1/16)*(25371-5*x)/(996+215*x)

This blog continues in the MRB constant N part 2.

Page 2:

Let x= MRB constant.  Each approximation is followed by a maple input so you can verify these approximations. 

 The digits of accuracy (D.O.A.) might seem embellished by 1 digit because it is computed by the reciprocal of the magnitude of error. For instance, The Champernowne Constant = 0.1234567891011121314151617181920212223243..., and its approximation constructed from the MRB constant is (1/2)*(8647+1490*x)/(31589+24300*x).  0.1234567891011121314151617181920212223243-(1/2)*(8647+1490*x)/(31589+24300*x)= 9.79861545008*10^(-19). Thus, according to this rule, there are 19 digits worth of accuracy in the approximation. However, You could argue that counting the 0, only 18 digits actually match .

 The first Foias constant-(24242/16437)*Pi+35/6-(1/15)*x  to all known digits of accuracy.  -(24242/16437)*Pi+35/6-(1/15)*x The figure eight knot hyperbolic volume = (1/4)*(-186121+6306*x)/(-22921+768*x)    to 18 digits of accuracy. (1/4)*(-186121+6306*x)/(-22921+768*x)

The second Foias constant (57063/157000)*exp(1)+777/3925+(63/7850)*x   to all known digits of accuracy.   (57063/157000)*exp(1)+777/3925+(63/7850)*x 

Viswanath's constant-(150/23177)*exp(1)+3800/3311+(100/9933)*x  to all known digits of accuracy.  -(150/23177)*exp(1)+3800/3311+(100/9933)*x

Viswanath's constant =  -1/9977*(4704*e+23744+1792*x) to all known digits of accuracy.  -1/9977*(4704*exp(1)+23744+1792*x)

Sarnak constant(1/4)*(29161-392*x)/(9960+473*x)  to 18 digits of accuracy.  (1/4)*(29161-392*x)/(9960+473*x)

Sarnak constant= (3/2)*(-159537+3418*x)/(-330691+7074*x)  to 18 digits of accuracy.  (3/2)*(-159537+3418*x)/(-330691+7074*x)

Plastic Constant = (1/3)*(192917+2712*x)/(48549+650*x) to 18 digits of accuracy.  (1/3)*(192917+2712*x)/(48549+650*x) 

 Rutherford Constant(2*(-4195+888*x))/(-10175+2036*x)  to 17 digits of accuracy.   (2*(-4195+888*x))/(-10175+2036*x)

Rutherford Constant =  (383657+33831*x)/(465607+45000*x) to 19 digits of accuracy.  (383657+33831*x)/(465607+45000*x)

Trotts first constant(6*(-492+19*x))/(-27097+344*x) to 17 digits of accuracy.   (6*(-492+19*x))/(-27097+344*x)

Trott's first  constant =  (1/4)*(3167-400*x)/(7099+165*x) to 17 digits of accuracy. (1/4)*(3167-400*x)/(7099+165*x)

Trott's first  constant =  (1/7)*(-2995+183*x)/(-3947+243*x)   to 17 digits of accuracy.  (1/7)*(-2995+183*x)/(-3947+243*x)

Sierpiński constant(6*(-16249+260*x))/(-118473+1822*x) to 18 digits of accuracy.   (6*(-16249+260*x))/(-118473+1822*x)

Sierpiński constant =   (-7652+81*x)/(-9550+1431*x)   to 17 digits of accuracy.   (-7652+81*x)/(-9550+1431*x)

 Mean tetrahedron-in-tetrahedron volume = (1/1004)*(202-279*x)/(8+3*x) to 18 digits of accuracy.   (1/1004)*(202-279*x)/(8+3*x)

Mean tetrahedron-in-tetrahedron volume = (2*(6377+75*x))/(739407-25150*x)    to 18 digits of accuracy.  (2*(6377+75*x))/(739407-25150*x)

The mean line - in - disk length(114341+2700*x)/(126239+3231*x)  to 19 digits of accuracy. (114341+2700*x)/(126239+3231*x)

The mean line-in-tesseract length =   (11/2)*(703-30*x)/(4927+27*x)  to 16 digits of accuracy. (11/2)*(703-30*x)/(4927+27*x)  The iterated exponential constant ( that is e^(-e) )(1/10)*(7169+417*x)/(10856+675*x)  to 18 digits of accuracy .  (1/10)*(7169+417*x)/(10856+675*x) 

 The iterated exponential constant =   (5239+576*x)/(79999+5504*x)   to 18 digits of accuracy.   (5239+576*x)/(79999+5504*x)

 The mean triangle - in - disk area (i.e. 0.0738800297...) =   (1798/79011)*Pi+188/79011+(4/79011)*x to 18 digits of accuracy.  (1798/79011)*Pi+188/79011+(4/79011)*x

 The mean triangle - in - disk area (i.e. 0.0738800297...) =  (1/4)*(75679+3252*x)/(256201+10400*x)    to 19 digits of accuracy.  (1/4)*(75679+3252*x)/(256201+10400*x)

 Nested Radical Constant =  (100/3)*(254+47*x)/(4790+1031*x)  to 18 digits of accuracy    (100/3)*(254+47*x)/(4790+1031*x),

 Ramanujan's first continued fraction constant =  (1/9)*(1658+305*x)/(122+367*x) to 18 digits of accuracy.(1/9)*(1658+305*x)/(122+367*x)

 Ramanujan's second continued fraction constant (i.e. 0.9999992087329…) =  -(1/60)*(-25537+54*x)/(420+29*x)  to 18 digits of accuracy.  -(1/60)*(-25537+54*x)/(420+29*x)

 Mertens Constant(1/2)*(50959+2614*x)/(97431+5030*x) to 19 digits of accuracy.  (1/2)*(50959+2614*x)/(97431+5030*x)

  

 

 

 

 

 

 

Let x=MRB constant and a= the constant that is the root of 45204 n^3-5015 n^2-22422 n+6145 between 0 and 0.5,

.

Pi*a = x^(1/150) with an error <29/100000000000000000000

 

restart; Digits := 25; x := .1878596424620671202485; a := fsolve(45204*n^3-5015*n^2-22422*n+6145, n, 0 .. .5); evalf(Pi*a-x^(1/150))

 

2.86304*10^(-20)

 

Let x=MRB constant and  h =The sixth hundred-dollar challenge constant ~=0.0619139544739909428481752164732

(7*h - 5008)/(49331*h - 8080) = x^(1/462) with an error < 20/1000000000000000000000

 

restart; Digits := 25; x := .1878596424620671202485; h := 0.619139544739909428481752164732e-1; a := (7*h-5008)/(49331*h-8080); a-x^(1/462)

1.91871*10^(-20)

 

 

Let x=MRB constant and  r=Fransén-Robinson constant ~=2.80777024202851936522150118655777293

4^x =  (4432-2001*r)/(33*r-1007) with an error < 2/100000000000000000.

restart; Digits := 25; x := .1878596424620671202485; r := 2.80777024202851936522150118655777293; a := (4432-2001*r)/(33*r-1007); 4^x-a

-1.9716872*10^(-17)

 

Let x=MRB constant and  v =hyperbolic volume of complement of the figure eight knot ~=2.029883212819307250042405108549.

(300 *(77-2*v))/(10873*v-39) = x^(1/245)  with an error < 20/100000000000000000000.

restart; Digits := 25; x := .1878596424620671202485; v := 2.029883212819307250042405108549; a := (300 (77-2v))/(10873v-39); x^(1/245)-a

1.900072*10^(-19)

 

 

marvinrayburns.com

Let x=MRB constant and a= the constant that is the real root of 53232 n^3-42020 n^2-4135 n-5430.

Pi*a = arcsinh(57*x) with an error < 34/2000000000000000000000

 

restart; Digits := 25; x := .1878596424620671202485; a := fsolve(53232*n^3-42020*n^2-4135*n-5430, n); evalf(Pi*a-arcsinh(57*x))

3.3929*10^(-20)

 

 

X=MRB constant:

Let c=Copeland-Erdős Constant ~=0.2357111317192329313741434753596

(74c+403)/(2271c-408) = log(27+x) with an error

restart; Digits := 20; x := .1878596424620671202485; c := .2357111317192329313741434753596; a := (74*c+403)/(2271*c-408); b := log(27+x); evalf(abs(a-b))

4.51*10^(-17)

 

 

Let u=Ubiquitous Constant ~=0.84721308479397908660649912348219

2020*u-273859/504*Pi = log(75+x) with an error < 4/10000000000000000

restart; Digits := 20; x := .1878596424620671202485; u := .84721308479397908660649912348219; a := 2020*u-(273859/504)*Pi; b := log(75+x); evalf(abs(a-b))

4.*10^(-16)

 

 Let v= Viswanath's constant = 1.1319882487943.

18*v-33931/6805*Pi = log(111+x) with an error < 12/1000000000000000

Restart: Digits := 20: x := .1878596424620671202485: v := 1.1319882487943: a := 18*v-33931/6805*P:

 b := log(111+x) : evalf(abs(a-b))

1.1178*10^(-14)

 

x=MRB constant:

Let p=Thue-Morse constant ~=0.412454033640107597783361368258

 

 (9071*p+640)/(734*p-260)= sinh(1/x) with an error <314/10000000000000000.

Digits := 24; x := .187859642462067120248517934054; p := .412454033640107597783361368258; a := sinh(1/x); b := (9071*p+640)/(734*p-260); a-b

3.1345962*10^(-14)

 

 Let m= Sarnak constant  ~= 0.7236484022982000094088491498

(1355-2047*m)/(78m-80) = log(213+x) with an error < 16/10000000000000000

Restart: Digits := 20: x := .1878596424620671202485: m := 0.7236484022982000094088491498: a := (1355-2047*m)/(78m-80): b := log(213+x) : evalf(abs(a-b))

1.6832*10^(-15)

 

Let 

 

A=10912090/5243631

and

B=2010579/4445410

 

Then

log(231+x)=     5.443230625222550370…

(A)^(3/4) pi^1= 5.443230625222548591…

(B)^(3/4) pi^2 =5.443230625222543779…


Let e= the number exp(1), x =MRB constant and s=Shallit Constant  =1.369451403937.

 (11315*s-821*exp(1))/8680= tan(22*x) with an error < 24/1000000000000000

Digits := 20; x := .1878596424620671202485; s := 1.369451403937; a := (11315*s-821*exp(1))*(1/8680); b := tan(22*x); evalf(a-b)

-2.320284*10^(-14)


Also,

 (2080*s)/267+(5*exp(1))/21 = -tan(59*x) with an error < 72/100000000000000


Digits := 20: x := .1878596424620671202485: s := 1.369451403937: a := 2080*s*(1/267)+5*exp(1)*(1/21): b := tan(59*x): evalf(abs(a+b))

7.1387576*10^(-13)


Also

(5127 *s-5291)/1320 = -tan(62*x) with an error < 16/100000000000000
Digits := 20; x := .1878596424620671202485; s := 1.369451403937; a := (5127*s-5291)*(1/1320); b := tan(62*x); evalf(abs(a+b))

1.535737*10^(-13)

4 5 6 7 8 9 10 Last Page 6 of 19