Powers of 2 - properties

//Federico Rampazzo, 2012
function magic(n, o, p, q, r, s, t){
  var ok = true;
  for (i=s; i<=t; i++){
    a = Math.pow(n,i);
    b = Math.pow(n,i-o)*p;
    c = Math.pow(n,i-q);
    d = Math.pow(n,i-r);
    e = a-b-c-d;
    console.log( i  + ' ' + a + '-' + b + '-' + c + '-' + d + '=' + e);
    if (e!=0) ok = false;
  }
  return ok;
}
// 2^x = (2^(x-10))*1000+2^(x-6)+2^(x-7) : True
magic(2, 10, 1000, 6, 7, -100, 100);

/*
-100 7.888609052210118e-31-7.703719777548943e-31-1.232595164407831e-32-6.162975822039155e-33=0
-99 1.5777218104420236e-30-1.5407439555097887e-30-2.465190328815662e-32-1.232595164407831e-32=0
-98 3.1554436208840472e-30-3.0814879110195774e-30-4.930380657631324e-32-2.465190328815662e-32=0
-97 6.310887241768095e-30-6.162975822039155e-30-9.860761315262648e-32-4.930380657631324e-32=0
-96 1.262177448353619e-29-1.232595164407831e-29-1.9721522630525295e-31-9.860761315262648e-32=0
-95 2.524354896707238e-29-2.465190328815662e-29-3.944304526105059e-31-1.9721522630525295e-31=0
-94 5.048709793414476e-29-4.930380657631324e-29-7.888609052210118e-31-3.944304526105059e-31=0
-93 1.0097419586828951e-28-9.860761315262648e-29-1.5777218104420236e-30-7.888609052210118e-31=0
-92 2.0194839173657902e-28-1.9721522630525295e-28-3.1554436208840472e-30-1.5777218104420236e-30=0
-91 4.0389678347315804e-28-3.944304526105059e-28-6.310887241768095e-30-3.1554436208840472e-30=0
-90 8.077935669463161e-28-7.888609052210118e-28-1.262177448353619e-29-6.310887241768095e-30=0
-89 1.6155871338926322e-27-1.5777218104420236e-27-2.524354896707238e-29-1.262177448353619e-29=0
-88 3.2311742677852644e-27-3.1554436208840472e-27-5.048709793414476e-29-2.524354896707238e-29=0
-87 6.462348535570529e-27-6.3108872417680944e-27-1.0097419586828951e-28-5.048709793414476e-29=0
-86 1.2924697071141057e-26-1.2621774483536189e-26-2.0194839173657902e-28-1.0097419586828951e-28=0
-85 2.5849394142282115e-26-2.5243548967072378e-26-4.0389678347315804e-28-2.0194839173657902e-28=0
-84 5.169878828456423e-26-5.0487097934144756e-26-8.077935669463161e-28-4.0389678347315804e-28=0
-83 1.0339757656912846e-25-1.0097419586828951e-25-1.6155871338926322e-27-8.077935669463161e-28=0
-82 2.0679515313825692e-25-2.0194839173657902e-25-3.2311742677852644e-27-1.6155871338926322e-27=0
-81 4.1359030627651384e-25-4.0389678347315804e-25-6.462348535570529e-27-3.2311742677852644e-27=0
-80 8.271806125530277e-25-8.077935669463161e-25-1.2924697071141057e-26-6.462348535570529e-27=0
-79 1.6543612251060553e-24-1.6155871338926322e-24-2.5849394142282115e-26-1.2924697071141057e-26=0
-78 3.308722450212111e-24-3.2311742677852644e-24-5.169878828456423e-26-2.5849394142282115e-26=0
-77 6.617444900424222e-24-6.462348535570529e-24-1.0339757656912846e-25-5.169878828456423e-26=0
-76 1.3234889800848443e-23-1.2924697071141057e-23-2.0679515313825692e-25-1.0339757656912846e-25=0
-75 2.6469779601696886e-23-2.5849394142282115e-23-4.1359030627651384e-25-2.0679515313825692e-25=0
-74 5.293955920339377e-23-5.169878828456423e-23-8.271806125530277e-25-4.1359030627651384e-25=0
-73 1.0587911840678754e-22-1.0339757656912846e-22-1.6543612251060553e-24-8.271806125530277e-25=0
-72 2.117582368135751e-22-2.0679515313825692e-22-3.308722450212111e-24-1.6543612251060553e-24=0
-71 4.235164736271502e-22-4.1359030627651384e-22-6.617444900424222e-24-3.308722450212111e-24=0
-70 8.470329472543003e-22-8.271806125530277e-22-1.3234889800848443e-23-6.617444900424222e-24=0
-69 1.6940658945086007e-21-1.6543612251060553e-21-2.6469779601696886e-23-1.3234889800848443e-23=0
-68 3.3881317890172014e-21-3.3087224502121107e-21-5.293955920339377e-23-2.6469779601696886e-23=0
-67 6.776263578034403e-21-6.6174449004242214e-21-1.0587911840678754e-22-5.293955920339377e-23=0
-66 1.3552527156068805e-20-1.3234889800848443e-20-2.117582368135751e-22-1.0587911840678754e-22=0
-65 2.710505431213761e-20-2.6469779601696886e-20-4.235164736271502e-22-2.117582368135751e-22=0
-64 5.421010862427522e-20-5.293955920339377e-20-8.470329472543003e-22-4.235164736271502e-22=0
-63 1.0842021724855044e-19-1.0587911840678754e-19-1.6940658945086007e-21-8.470329472543003e-22=0
-62 2.168404344971009e-19-2.1175823681357508e-19-3.3881317890172014e-21-1.6940658945086007e-21=0
-61 4.336808689942018e-19-4.2351647362715017e-19-6.776263578034403e-21-3.3881317890172014e-21=0
-60 8.673617379884035e-19-8.470329472543003e-19-1.3552527156068805e-20-6.776263578034403e-21=0
-59 1.734723475976807e-18-1.6940658945086007e-18-2.710505431213761e-20-1.3552527156068805e-20=0
-58 3.469446951953614e-18-3.3881317890172014e-18-5.421010862427522e-20-2.710505431213761e-20=0
-57 6.938893903907228e-18-6.776263578034403e-18-1.0842021724855044e-19-5.421010862427522e-20=0
-56 1.3877787807814457e-17-1.3552527156068805e-17-2.168404344971009e-19-1.0842021724855044e-19=0
-55 2.7755575615628914e-17-2.710505431213761e-17-4.336808689942018e-19-2.168404344971009e-19=0
-54 5.551115123125783e-17-5.421010862427522e-17-8.673617379884035e-19-4.336808689942018e-19=0
-53 1.1102230246251565e-16-1.0842021724855044e-16-1.734723475976807e-18-8.673617379884035e-19=0
-52 2.220446049250313e-16-2.1684043449710089e-16-3.469446951953614e-18-1.734723475976807e-18=0
-51 4.440892098500626e-16-4.3368086899420177e-16-6.938893903907228e-18-3.469446951953614e-18=0
-50 8.881784197001252e-16-8.673617379884035e-16-1.3877787807814457e-17-6.938893903907228e-18=0
-49 1.7763568394002505e-15-1.734723475976807e-15-2.7755575615628914e-17-1.3877787807814457e-17=0
-48 3.552713678800501e-15-3.469446951953614e-15-5.551115123125783e-17-2.7755575615628914e-17=0
-47 7.105427357601002e-15-6.938893903907228e-15-1.1102230246251565e-16-5.551115123125783e-17=0
-46 1.4210854715202004e-14-1.3877787807814457e-14-2.220446049250313e-16-1.1102230246251565e-16=0
-45 2.842170943040401e-14-2.7755575615628914e-14-4.440892098500626e-16-2.220446049250313e-16=0
-44 5.684341886080802e-14-5.551115123125783e-14-8.881784197001252e-16-4.440892098500626e-16=0
-43 1.1368683772161603e-13-1.1102230246251565e-13-1.7763568394002505e-15-8.881784197001252e-16=0
-42 2.2737367544323206e-13-2.220446049250313e-13-3.552713678800501e-15-1.7763568394002505e-15=0
-41 4.547473508864641e-13-4.440892098500626e-13-7.105427357601002e-15-3.552713678800501e-15=0
-40 9.094947017729282e-13-8.881784197001252e-13-1.4210854715202004e-14-7.105427357601002e-15=0
-39 1.8189894035458565e-12-1.7763568394002505e-12-2.842170943040401e-14-1.4210854715202004e-14=0
-38 3.637978807091713e-12-3.552713678800501e-12-5.684341886080802e-14-2.842170943040401e-14=0
-37 7.275957614183426e-12-7.105427357601002e-12-1.1368683772161603e-13-5.684341886080802e-14=0
-36 1.4551915228366852e-11-1.4210854715202004e-11-2.2737367544323206e-13-1.1368683772161603e-13=0
-35 2.9103830456733704e-11-2.8421709430404007e-11-4.547473508864641e-13-2.2737367544323206e-13=0
-34 5.820766091346741e-11-5.6843418860808015e-11-9.094947017729282e-13-4.547473508864641e-13=0
-33 1.1641532182693481e-10-1.1368683772161603e-10-1.8189894035458565e-12-9.094947017729282e-13=0
-32 2.3283064365386963e-10-2.2737367544323206e-10-3.637978807091713e-12-1.8189894035458565e-12=0
-31 4.656612873077393e-10-4.547473508864641e-10-7.275957614183426e-12-3.637978807091713e-12=0
-30 9.313225746154785e-10-9.094947017729282e-10-1.4551915228366852e-11-7.275957614183426e-12=0
-29 1.862645149230957e-9-1.8189894035458565e-9-2.9103830456733704e-11-1.4551915228366852e-11=0
-28 3.725290298461914e-9-3.637978807091713e-9-5.820766091346741e-11-2.9103830456733704e-11=0
-27 7.450580596923828e-9-7.275957614183426e-9-1.1641532182693481e-10-5.820766091346741e-11=0
-26 1.4901161193847656e-8-1.4551915228366852e-8-2.3283064365386963e-10-1.1641532182693481e-10=0
-25 2.9802322387695312e-8-2.9103830456733704e-8-4.656612873077393e-10-2.3283064365386963e-10=0
-24 5.960464477539063e-8-5.820766091346741e-8-9.313225746154785e-10-4.656612873077393e-10=0
-23 1.1920928955078125e-7-1.1641532182693481e-7-1.862645149230957e-9-9.313225746154785e-10=0
-22 2.384185791015625e-7-2.3283064365386963e-7-3.725290298461914e-9-1.862645149230957e-9=0
-21 4.76837158203125e-7-4.6566128730773926e-7-7.450580596923828e-9-3.725290298461914e-9=0
-20 9.5367431640625e-7-9.313225746154785e-7-1.4901161193847656e-8-7.450580596923828e-9=0
-19 0.0000019073486328125-0.000001862645149230957-2.9802322387695312e-8-1.4901161193847656e-8=0
-18 0.000003814697265625-0.000003725290298461914-5.960464477539063e-8-2.9802322387695312e-8=0
-17 0.00000762939453125-0.000007450580596923828-1.1920928955078125e-7-5.960464477539063e-8=0
-16 0.0000152587890625-0.000014901161193847656-2.384185791015625e-7-1.1920928955078125e-7=0
-15 0.000030517578125-0.000029802322387695312-4.76837158203125e-7-2.384185791015625e-7=0
-14 0.00006103515625-0.000059604644775390625-9.5367431640625e-7-4.76837158203125e-7=0
-13 0.0001220703125-0.00011920928955078125-0.0000019073486328125-9.5367431640625e-7=0
-12 0.000244140625-0.0002384185791015625-0.000003814697265625-0.0000019073486328125=0
-11 0.00048828125-0.000476837158203125-0.00000762939453125-0.000003814697265625=0
-10 0.0009765625-0.00095367431640625-0.0000152587890625-0.00000762939453125=0
-9 0.001953125-0.0019073486328125-0.000030517578125-0.0000152587890625=0
-8 0.00390625-0.003814697265625-0.00006103515625-0.000030517578125=0
-7 0.0078125-0.00762939453125-0.0001220703125-0.00006103515625=0
-6 0.015625-0.0152587890625-0.000244140625-0.0001220703125=0
-5 0.03125-0.030517578125-0.00048828125-0.000244140625=0
-4 0.0625-0.06103515625-0.0009765625-0.00048828125=0
-3 0.125-0.1220703125-0.001953125-0.0009765625=0
-2 0.25-0.244140625-0.00390625-0.001953125=0
-1 0.5-0.48828125-0.0078125-0.00390625=0
0 1-0.9765625-0.015625-0.0078125=0
1 2-1.953125-0.03125-0.015625=0
2 4-3.90625-0.0625-0.03125=0
3 8-7.8125-0.125-0.0625=0
4 16-15.625-0.25-0.125=0
5 32-31.25-0.5-0.25=0
6 64-62.5-1-0.5=0
7 128-125-2-1=0
8 256-250-4-2=0
9 512-500-8-4=0
10 1024-1000-16-8=0
11 2048-2000-32-16=0
12 4096-4000-64-32=0
13 8192-8000-128-64=0
14 16384-16000-256-128=0
15 32768-32000-512-256=0
16 65536-64000-1024-512=0
17 131072-128000-2048-1024=0
18 262144-256000-4096-2048=0
19 524288-512000-8192-4096=0
20 1048576-1024000-16384-8192=0
21 2097152-2048000-32768-16384=0
22 4194304-4096000-65536-32768=0
23 8388608-8192000-131072-65536=0
24 16777216-16384000-262144-131072=0
25 33554432-32768000-524288-262144=0
26 67108864-65536000-1048576-524288=0
27 134217728-131072000-2097152-1048576=0
28 268435456-262144000-4194304-2097152=0
29 536870912-524288000-8388608-4194304=0
30 1073741824-1048576000-16777216-8388608=0
31 2147483648-2097152000-33554432-16777216=0
32 4294967296-4194304000-67108864-33554432=0
33 8589934592-8388608000-134217728-67108864=0
34 17179869184-16777216000-268435456-134217728=0
35 34359738368-33554432000-536870912-268435456=0
36 68719476736-67108864000-1073741824-536870912=0
37 137438953472-134217728000-2147483648-1073741824=0
38 274877906944-268435456000-4294967296-2147483648=0
39 549755813888-536870912000-8589934592-4294967296=0
40 1099511627776-1073741824000-17179869184-8589934592=0
41 2199023255552-2147483648000-34359738368-17179869184=0
42 4398046511104-4294967296000-68719476736-34359738368=0
43 8796093022208-8589934592000-137438953472-68719476736=0
44 17592186044416-17179869184000-274877906944-137438953472=0
45 35184372088832-34359738368000-549755813888-274877906944=0
46 70368744177664-68719476736000-1099511627776-549755813888=0
47 140737488355328-137438953472000-2199023255552-1099511627776=0
48 281474976710656-274877906944000-4398046511104-2199023255552=0
49 562949953421312-549755813888000-8796093022208-4398046511104=0
50 1125899906842624-1099511627776000-17592186044416-8796093022208=0
51 2251799813685248-2199023255552000-35184372088832-17592186044416=0
52 4503599627370496-4398046511104000-70368744177664-35184372088832=0
53 9007199254740992-8796093022208000-140737488355328-70368744177664=0
54 18014398509481984-17592186044416000-281474976710656-140737488355328=0
55 36028797018963970-35184372088832000-562949953421312-281474976710656=0
56 72057594037927940-70368744177664000-1125899906842624-562949953421312=0
57 144115188075855870-140737488355328000-2251799813685248-1125899906842624=0
58 288230376151711740-281474976710656000-4503599627370496-2251799813685248=0
59 576460752303423500-562949953421312000-9007199254740992-4503599627370496=0
60 1152921504606847000-1125899906842624000-18014398509481984-9007199254740992=0
61 2305843009213694000-2251799813685248000-36028797018963970-18014398509481984=0
62 4611686018427388000-4503599627370496000-72057594037927940-36028797018963970=0
63 9223372036854776000-9007199254740992000-144115188075855870-72057594037927940=0
64 18446744073709552000-18014398509481984000-288230376151711740-144115188075855870=0
65 36893488147419103000-36028797018963970000-576460752303423500-288230376151711740=0
66 73786976294838210000-72057594037927940000-1152921504606847000-576460752303423500=0
67 147573952589676410000-144115188075855870000-2305843009213694000-1152921504606847000=0
68 295147905179352830000-288230376151711740000-4611686018427388000-2305843009213694000=0
69 590295810358705700000-576460752303423500000-9223372036854776000-4611686018427388000=0
70 1.1805916207174113e+21-1.152921504606847e+21-18446744073709552000-9223372036854776000=0
71 2.3611832414348226e+21-2.305843009213694e+21-36893488147419103000-18446744073709552000=0
72 4.722366482869645e+21-4.611686018427388e+21-73786976294838210000-36893488147419103000=0
73 9.44473296573929e+21-9.223372036854776e+21-147573952589676410000-73786976294838210000=0
74 1.888946593147858e+22-1.8446744073709552e+22-295147905179352830000-147573952589676410000=0
75 3.777893186295716e+22-3.6893488147419103e+22-590295810358705700000-295147905179352830000=0
76 7.555786372591432e+22-7.378697629483821e+22-1.1805916207174113e+21-590295810358705700000=0
77 1.5111572745182865e+23-1.4757395258967641e+23-2.3611832414348226e+21-1.1805916207174113e+21=0
78 3.022314549036573e+23-2.9514790517935283e+23-4.722366482869645e+21-2.3611832414348226e+21=0
79 6.044629098073146e+23-5.9029581035870565e+23-9.44473296573929e+21-4.722366482869645e+21=0
80 1.2089258196146292e+24-1.1805916207174113e+24-1.888946593147858e+22-9.44473296573929e+21=0
81 2.4178516392292583e+24-2.3611832414348226e+24-3.777893186295716e+22-1.888946593147858e+22=0
82 4.835703278458517e+24-4.722366482869645e+24-7.555786372591432e+22-3.777893186295716e+22=0
83 9.671406556917033e+24-9.44473296573929e+24-1.5111572745182865e+23-7.555786372591432e+22=0
84 1.9342813113834067e+25-1.888946593147858e+25-3.022314549036573e+23-1.5111572745182865e+23=0
85 3.8685626227668134e+25-3.777893186295716e+25-6.044629098073146e+23-3.022314549036573e+23=0
86 7.737125245533627e+25-7.555786372591432e+25-1.2089258196146292e+24-6.044629098073146e+23=0
87 1.5474250491067253e+26-1.5111572745182865e+26-2.4178516392292583e+24-1.2089258196146292e+24=0
88 3.094850098213451e+26-3.022314549036573e+26-4.835703278458517e+24-2.4178516392292583e+24=0
89 6.189700196426902e+26-6.044629098073146e+26-9.671406556917033e+24-4.835703278458517e+24=0
90 1.2379400392853803e+27-1.2089258196146292e+27-1.9342813113834067e+25-9.671406556917033e+24=0
91 2.4758800785707605e+27-2.4178516392292583e+27-3.8685626227668134e+25-1.9342813113834067e+25=0
92 4.951760157141521e+27-4.8357032784585167e+27-7.737125245533627e+25-3.8685626227668134e+25=0
93 9.903520314283042e+27-9.671406556917033e+27-1.5474250491067253e+26-7.737125245533627e+25=0
94 1.9807040628566084e+28-1.9342813113834067e+28-3.094850098213451e+26-1.5474250491067253e+26=0
95 3.961408125713217e+28-3.8685626227668134e+28-6.189700196426902e+26-3.094850098213451e+26=0
96 7.922816251426434e+28-7.737125245533627e+28-1.2379400392853803e+27-6.189700196426902e+26=0
97 1.5845632502852868e+29-1.5474250491067253e+29-2.4758800785707605e+27-1.2379400392853803e+27=0
98 3.1691265005705735e+29-3.0948500982134507e+29-4.951760157141521e+27-2.4758800785707605e+27=0
99 6.338253001141147e+29-6.1897001964269014e+29-9.903520314283042e+27-4.951760157141521e+27=0
100 1.2676506002282294e+30-1.2379400392853803e+30-1.9807040628566084e+28-9.903520314283042e+27=0
*/

/*
3^x = (3^(x-o))*p+3^(x-q)+3^(x-r)
3^x = 3^x *(3^(-o)*p+3^(-q)+3^(-r))

3^(-o)*p+3^(-q)+3^(-r)=1
r = -log3(-3^(-o)*p-3^(-q)+1))
q = -log3(-3^(-o)*p-3^(-r)+1))

*/