How many
integer values of n satisfy this equation?
(6+n)n + (7+n)n = 12n + 1
Prove that there are no others.
The solutions are n = 0, 2 and 3.
We are seeking a difference of zero between the left hand side and the right hand side of the equation.
The following is a list of n, the LHS, the RHS and the difference:
n LHS RHS Difference
-19 -3.81411821865717e-21 1 -1
-18 2.17419826667822e-19 1 -1
-17 -1.19784466890014e-17 1 -1
-16 6.39659527735429e-16 1 -0.999999999999999
-15 -3.32786451800229e-14 1 -1.00000000000003
-14 1.7018148144026e-12 1 -0.999999999998299
-13 -8.68866976390128e-11 1.00000000000001 -1.0000000000869
-12 4.55539365799778e-09 1.00000000000011 -0.999999995444719
-11 -2.58898579101562e-07 1.00000000000135 -1.00000025889992
-10 1.78887621248365e-05 1.00000000001615 -0.999982111254026
-9 -0.00200393026342529 1.00000000019381 -1.00200393045723
In the above portion, the farther you go in the negative direction the difference between the LHS and RHS approaches 1 (negative when RHS is subtracted from LHS) as the LHS gets tinier and tinier and the RHS differs from 1 by a tinier and tinier amount, with no zeros.
In the portion below, the numbers (and infinities) speak for themselves: there are no negative values for n that work:
-8 1.00390625 1.00000000232568 0.00390624767431969
-7 Inf 1.00000002790816 Inf
-6 Inf 1.00000033489798 Inf
-5 1.03125 1.00000401877572 0.0312459812242798
-4 0.0748456790123457 1.00004822530864 -0.925202546296296
-3 0.052662037037037 1.0005787037037 -0.947916666666667
-2 0.1025 1.00694444444444 -0.904444444444444
-1 0.366666666666667 1.08333333333333 -0.716666666666667
The solutions, n = 0, 2 and 3, lie within the next portion:
0 2 2 0
1 15 13 2
2 145 145 0
3 1729 1729 0
4 24641 20737 3904
5 409883 248833 161050
6 7812793 2985985 4826808
7 168162021 35831809 132330212
After that, since the LHS increases the base as well as the exponent(s), it increases at a much faster rate, and so prevents a solution from happening thereafter:
8 4038679681 429981697 3608697984
9 107162836111 5159780353 102003055758
10 3115505528225 61917364225 3053588164000
11 98540306386865 743008370689 97797298016176
12 3.37014630049234e+15 8916100448257 3.36123020004408e+15
13 1.23972983462257e+17 106993205379073 1.23865990256878e+17
14 4.88231993252151e+18 1.28391846454886e+15 4.88103601405696e+18
15 2.05002386598364e+20 1.54070215745864e+16 2.04986979576789e+20
16 9.14397191202006e+21 1.84884258895036e+17 9.14378702776117e+21
17 4.31847834543346e+23 2.21861110674044e+18 4.31845615932239e+23
18 2.15310623079512e+25 2.66233332808852e+19 2.1531035684618e+25
19 1.13026514590953e+27 3.19479999370623e+20 1.13026482642953e+27
20 6.23193071704256e+28 3.83375999244748e+21 6.23193033366656e+28
21 3.60107196225389e+30 4.60051199093697e+22 3.60107191624877e+30
22 2.17634737830129e+32 5.52061438912436e+23 2.17634737278067e+32
23 1.37310386004494e+34 6.62473726694924e+24 1.37310385938247e+34
24 9.02842197446528e+35 7.94968472033908e+25 9.02842197367031e+35
25 6.17680883550487e+37 9.5396216644069e+26 6.17680883540947e+37
26 4.39057003540788e+39 1.14475459972883e+28 4.39057003539643e+39
27 3.23807963356253e+41 1.37370551967459e+29 3.23807963356116e+41
28 2.47462722572583e+43 1.64844662360951e+30 2.47462722572567e+43
29 1.95735634983488e+45 1.97813594833142e+31 1.95735634983486e+45
30 1.60060091591683e+47 2.3737631379977e+32 1.60060091591683e+47
31 1.35174398771445e+49 2.84851576559724e+33 1.35174398771445e+49
32 1.17781353277787e+51 3.41821891871669e+34 1.17781353277787e+51
33 1.05785653979192e+53 4.10186270246002e+35 1.05785653979192e+53
34 9.78510382475185e+54 4.92223524295203e+36 9.78510382475185e+54
35 9.31394767964079e+56 5.90668229154243e+37 9.31394767964079e+56
36 9.11575017172324e+58 7.08801874985092e+38 9.11575017172324e+58
37 9.1668404863761e+60 8.5056224998211e+39 9.1668404863761e+60
38 9.46477733303434e+62 1.02067469997853e+41 9.46477733303434e+62
39 1.00270840799131e+65 1.22480963997424e+42 1.00270840799131e+65
40 1.0892727333518e+67 1.46977156796909e+43 1.0892727333518e+67
41 1.2126449235799e+69 1.7637258815629e+44 1.2126449235799e+69
42 1.38265957782068e+71 2.11647105787548e+45 1.38265957782068e+71
43 1.613772900095e+73 2.53976526945058e+46 1.613772900095e+73
44 1.92702210736814e+75 3.0477183233407e+47 1.92702210736814e+75
45 2.35304181807201e+77 3.65726198800884e+48 2.35304181807201e+77
46 2.93671720695092e+79 4.3887143856106e+49 2.93671720695092e+79
47 3.74441305656253e+81 5.26645726273273e+50 3.74441305656253e+81
48 4.87531106782119e+83 6.31974871527927e+51 4.87531106782119e+83
49 6.47937928474161e+85 7.58369845833512e+52 6.47937928474161e+85
50 8.7861733736465e+87 9.10043815000215e+53 8.7861733736465e+87
51 1.21515331975502e+90 1.09205257800026e+55 1.21515331975502e+90
52 1.71341845821197e+92 1.31046309360031e+56 1.71341845821197e+92
53 2.46228927464431e+94 1.57255571232037e+57 2.46228927464431e+94
54 3.60500573079924e+96 1.88706685478445e+58 3.60500573079924e+96
55 5.37548366118061e+98 2.26448022574133e+59 5.37548366118061e+98
56 8.16079474264152e+100 2.7173762708896e+60 8.16079474264152e+100
57 1.26099746468469e+103 3.26085152506752e+61 1.26099746468469e+103
58 1.98258130837853e+105 3.91302183008103e+62 1.98258130837853e+105
59 3.17070214382941e+107 4.69562619609723e+63 3.17070214382941e+107
60 5.15661070776904e+109 5.63475143531668e+64 5.15661070776904e+109
61 8.52586644940379e+111 6.76170172238001e+65 8.52586644940379e+111
62 1.43272426027499e+114 8.11404206685602e+66 1.43272426027499e+114
63 2.44638540556325e+116 9.73685048022722e+67 2.44638540556325e+116
64 4.24342997956383e+118 1.16842205762727e+69 4.24342997956383e+118
65 7.47539488055743e+120 1.40210646915272e+70 7.47539488055743e+120
66 1.33713225608709e+123 1.68252776298326e+71 1.33713225608709e+123
67 2.42794549042308e+125 2.01903331557992e+72 2.42794549042308e+125
68 4.4743704412526e+127 2.4228399786959e+73 4.4743704412526e+127
69 8.36681646086608e+129 2.90740797443508e+74 8.36681646086608e+129
70 1.58720620982603e+132 3.4888895693221e+75 1.58720620982603e+132
71 3.05395979129668e+134 4.18666748318651e+76 3.05395979129668e+134
72 5.95887941845602e+136 5.02400097982382e+77 5.95887941845602e+136
73 1.17883767069465e+139 6.02880117578858e+78 1.17883767069465e+139
74 2.36401692666134e+141 7.2345614109463e+79 2.36401692666134e+141
75 4.80480223782581e+143 8.68147369313556e+80 4.80480223782581e+143
76 9.89582639680142e+145 1.04177684317627e+82 9.89582639680142e+145
77 2.0649306732849e+148 1.25013221181152e+83 2.0649306732849e+148
78 4.36478535272934e+150 1.50015865417382e+84 4.36478535272934e+150
79 9.34444051508666e+152 1.80019038500859e+85 9.34444051508666e+152
80 2.02585086334201e+155 2.16022846201031e+86 2.02585086334201e+155
81 4.44690502934438e+157 2.59227415441237e+87 4.44690502934438e+157
82 9.88185686372318e+159 3.11072898529484e+88 9.88185686372318e+159
83 2.22272263222243e+162 3.73287478235381e+89 2.22272263222243e+162
84 5.05982162659218e+164 4.47944973882457e+90 5.05982162659218e+164
85 1.16553965236007e+167 5.37533968658949e+91 1.16553965236007e+167
86 2.71644502537582e+169 6.45040762390738e+92 2.71644502537582e+169
87 6.40469263971301e+171 7.74048914868886e+93 6.40469263971301e+171
88 1.52743234070344e+174 9.28858697842663e+94 1.52743234070344e+174
89 3.68413962305666e+176 1.1146304374112e+96 3.68413962305666e+176
90 8.9859898568642e+178 1.33755652489344e+97 8.9859898568642e+178
91 2.21614434669223e+181 1.60506782987212e+98 2.21614434669223e+181
92 5.5256088123209e+183 1.92608139584655e+99 5.5256088123209e+183
93 1.39271102835781e+186 2.31129767501586e+100 1.39271102835781e+186
94 3.54805698025623e+188 2.77355721001903e+101 3.54805698025623e+188
95 9.13523674567464e+190 3.32826865202283e+102 9.13523674567464e+190
96 2.37684387731772e+193 3.9939223824274e+103 2.37684387731772e+193
97 6.24864857921372e+195 4.79270685891288e+104 6.24864857921372e+195
98 1.65970180952816e+198 5.75124823069546e+105 1.65970180952816e+198
99 4.453355983893e+200 6.90149787683455e+106 4.453355983893e+200
100 1.2070184090809e+203 8.28179745220145e+107 1.2070184090809e+203
clc, clearvars
for n=-50:100
lhs= (6+n)^n+(7+n)^n;
rhs= 12^n+1;
disp([n lhs rhs lhs-rhs]);
end
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Posted by Charlie
on 2021-11-22 11:56:13 |
computer solution
