Determine the total number of ordered pairs(x,y) of positive integers with 2≤ x,y ≤ 2023 that satisfy this equation:
xlogy(x-4) = ylogx(yx-3)
clearvars;clc
syms x y
fimplicit(x^(log(x^-4)/log(y))==y^(log(y*x^-3)/log(x)))
grid
ct=0;
for x=2:2023
for y=2:2023
v1=x^(log(x^-4)/log(y));
v2=y^(log(y*x^-3)/log(x));
if abs(v1-v2)<abs(v1/10000000)
ct=ct+1;
disp([x y v1 v2])
end
end
end
based on solution by Wolfram Alpha, and shows the table:
x y LHS RHS
2 4 0.25 0.25
3 9 0.111111111111111 0.111111111111111
4 16 0.0625 0.0625
5 25 0.04 0.04
6 36 0.0277777777777778 0.0277777777777778
7 49 0.0204081632653061 0.0204081632653061
8 64 0.015625 0.015625
9 81 0.0123456790123457 0.0123456790123457
10 100 0.01 0.01
11 121 0.00826446280991736 0.00826446280991736
12 144 0.00694444444444444 0.00694444444444444
13 169 0.00591715976331361 0.00591715976331361
14 196 0.00510204081632654 0.00510204081632652
15 225 0.00444444444444444 0.00444444444444444
16 256 0.00390625 0.00390625
17 289 0.00346020761245675 0.00346020761245675
18 324 0.00308641975308642 0.00308641975308642
19 361 0.00277008310249307 0.00277008310249307
20 400 0.0025 0.0025
21 441 0.00226757369614512 0.00226757369614512
22 484 0.00206611570247934 0.00206611570247934
23 529 0.00189035916824197 0.00189035916824197
24 576 0.00173611111111111 0.00173611111111111
25 625 0.0016 0.0016
26 676 0.0014792899408284 0.0014792899408284
27 729 0.00137174211248285 0.00137174211248285
28 784 0.00127551020408163 0.00127551020408163
29 841 0.00118906064209275 0.00118906064209275
30 900 0.00111111111111111 0.00111111111111111
31 961 0.00104058272632674 0.00104058272632674
32 1024 0.0009765625 0.0009765625
33 1089 0.000918273645546373 0.000918273645546373
34 1156 0.000865051903114187 0.000865051903114187
35 1225 0.000816326530612245 0.000816326530612245
36 1296 0.000771604938271605 0.000771604938271605
37 1369 0.000730460189919649 0.000730460189919649
38 1444 0.000692520775623269 0.000692520775623269
39 1521 0.000657462195923734 0.000657462195923734
40 1600 0.000625 0.000625
41 1681 0.000594883997620464 0.000594883997620464
42 1764 0.000566893424036281 0.000566893424036282
43 1849 0.000540832882639265 0.000540832882639265
44 1936 0.000516528925619835 0.000516528925619835
The curve is y=x^2, but strangely does not appear on the graph produced by fimplicit. That graph shows only the other solution given by Wolfram Alpha: y=1/x, but also other solutions in which either y or x is very close to but not equal to 1 (on the higher-than-1 side).
The LHS and RHS are each equal to 1/y = 1/x^2.
After x=44, the y values go beyond the requested high limit.
Therefore there are 43 values in the range specified that are integers.
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Posted by Charlie
on 2023-07-30 09:43:47 |
computer solution
