I have this equation which I need to prove or disprove-
$$\frac{z^3}{x^3+y^3+2xyz}=3$$$(x,y,z\in \mathbb N)$
I understand that for this equation to be true $z^3$ must be of the form $3k$ and $x^3+y^3+2xyz$ must be of the form $k$.
I suspect this equation has no solution but I cannot prove it.Am I on the correct line?
Thanks for any help!!
Sigh. It seems possible to get the ratio an integer when $x,y,z$ are positive integers, but the smallest possible ratio is $5$ rather than $3.$ Also, usually only one triple with $\gcd(x,y,z) = 1$ for each ratio, I did eventually get a repeat for ratio $49.$ Strong preference for small prime factors, and few of them, in the ratios. Indeed, very few prime ratios, just $5$ and $13$ so far. Interesting that $3$ seems to be impossible when $9$ and $27$ work. This is $x,y,z \leq 1200.$
5 x: 4 y: 51 z: 95 ratio: 5 = 5
9 x: 4 y: 5 z: 21 ratio: 9 = 3^2
13 x: 1 y: 4 z: 13 ratio: 13 = 13
27 x: 19 y: 21 z: 156 ratio: 27 = 3^3
32 x: 3 y: 29 z: 112 ratio: 32 = 2^5
38 x: 39 y: 265 z: 1178 ratio: 38 = 2 19
49 x: 19 y: 32 z: 259 ratio: 49 = 7^2
49 x: 24 y: 71 z: 455 ratio: 49 = 7^2
72 x: 1 y: 2 z: 18 ratio: 72 = 2^3 3^2
96 x: 1 y: 5 z: 36 ratio: 96 = 2^5 3
112 x: 1 y: 3 z: 28 ratio: 112 = 2^4 7
128 x: 1 y: 11 z: 72 ratio: 128 = 2^7
134 x: 7 y: 9 z: 134 ratio: 134 = 2 67
152 x: 27 y: 58 z: 722 ratio: 152 = 2^3 19
169 x: 7 y: 132 z: 871 ratio: 169 = 13^2
248 x: 1 y: 6 z: 62 ratio: 248 = 2^3 31
256 x: 1 y: 5 z: 56 ratio: 256 = 2^8
362 x: 3 y: 37 z: 362 ratio: 362 = 2 181
392 x: 2 y: 3 z: 70 ratio: 392 = 2^3 7^2
432 x: 5 y: 43 z: 504 ratio: 432 = 2^4 3^3
512 x: 9 y: 31 z: 560 ratio: 512 = 2^9
837 x: 4 y: 11 z: 279 ratio: 837 = 3^3 31
1000 x: 13 y: 14 z: 610 ratio: 1000 = 2^3 5^3
1352 x: 3 y: 4 z: 182 ratio: 1352 = 2^3 13^2
1352 x: 7 y: 41 z: 936 ratio: 1352 = 2^3 13^2
1369 x: 1 y: 36 z: 481 ratio: 1369 = 37^2
1400 x: 1 y: 24 z: 350 ratio: 1400 = 2^3 5^2 7
1539 x: 1 y: 8 z: 171 ratio: 1539 = 3^4 19
1600 x: 2 y: 3 z: 140 ratio: 1600 = 2^6 5^2
1849 x: 1 y: 7 z: 172 ratio: 1849 = 43^2
2072 x: 15 y: 17 z: 1036 ratio: 2072 = 2^3 7 37
2457 x: 1 y: 12 z: 273 ratio: 2457 = 3^3 7 13
2646 x: 5 y: 11 z: 546 ratio: 2646 = 2 3^3 7^2
3042 x: 1 y: 23 z: 468 ratio: 3042 = 2 3^2 13^2
3528 x: 4 y: 5 z: 378 ratio: 3528 = 2^3 3^2 7^2
4563 x: 1 y: 4 z: 195 ratio: 4563 = 3^3 13^2
4732 x: 5 y: 11 z: 728 ratio: 4732 = 2^2 7 13^2
4750 x: 3 y: 5 z: 380 ratio: 4750 = 2 5^3 19
5616 x: 3 y: 17 z: 780 ratio: 5616 = 2^4 3^3 13
5824 x: 1 y: 10 z: 364 ratio: 5824 = 2^6 7 13
5929 x: 3 y: 8 z: 539 ratio: 5929 = 7^2 11^2
6048 x: 1 y: 5 z: 252 ratio: 6048 = 2^5 3^3 7
7688 x: 5 y: 6 z: 682 ratio: 7688 = 2^3 31^2
8000 x: 1 y: 2 z: 180 ratio: 8000 = 2^6 5^3
8192 x: 1 y: 3 z: 224 ratio: 8192 = 2^13
8232 x: 1 y: 8 z: 378 ratio: 8232 = 2^3 3 7^3
9317 x: 1 y: 20 z: 693 ratio: 9317 = 7 11^3
14112 x: 1 y: 17 z: 756 ratio: 14112 = 2^5 3^2 7^2
14792 x: 6 y: 7 z: 1118 ratio: 14792 = 2^3 43^2
16129 x: 1 y: 20 z: 889 ratio: 16129 = 127^2
16625 x: 1 y: 12 z: 665 ratio: 16625 = 5^3 7 19
16807 x: 3 y: 4 z: 637 ratio: 16807 = 7^5
33280 x: 1 y: 4 z: 520 ratio: 33280 = 2^9 5 13
39528 x: 1 y: 14 z: 1098 ratio: 39528 = 2^3 3^4 61
61504 x: 1 y: 6 z: 868 ratio: 61504 = 2^6 31^2
74304 x: 1 y: 7 z: 1032 ratio: 74304 = 2^6 3^3 43
100352 x: 1 y: 5 z: 1008 ratio: 100352 = 2^11 7^2
The denominator is nonzero with all positive variables, but there are plenty of ways to get zero with mixed signs, here are some with $\gcd(x,y,z) = 1.$
x: -1 y: 1 z: 0
x: -1 y: -1 z: 1
x: 0 y: 0 z: 1
x: -4 y: 8 z: 7
x: -8 y: -4 z: 9
x: -3 y: 9 z: 13
x: -9 y: -3 z: 14
x: -24 y: 36 z: 19
x: -36 y: -24 z: 35
x: -72 y: 96 z: 37
x: -45 y: 75 z: 49
x: -160 y: 200 z: 61
x: -5 y: 25 z: 62
x: -25 y: -5 z: 63
x: -8 y: 32 z: 63
x: -32 y: -8 z: 65
x: -75 y: -45 z: 76
x: -300 y: 360 z: 91
x: -96 y: -72 z: 91
x: -175 y: 245 z: 109
x: -40 y: 100 z: 117
x: -504 y: 588 z: 127
x: -100 y: -40 z: 133
x: -63 y: 147 z: 158
x: -784 y: 896 z: 169
x: -7 y: 49 z: 171
x: -49 y: -7 z: 172
x: -147 y: -63 z: 185
x: -200 y: -160 z: 189
x: -441 y: 567 z: 193
x: -12 y: 72 z: 215
x: -72 y: -12 z: 217
Allowing all signs, here are the ratios with $xyz \neq 0$ and denominator not zero, from $-200$ to $200,$ with $|x|, |y|, |z| \leq 600.$
-200 x: -1 y: 1 z: 20 ratio: -200 = -1 * 2^3 5^2
-200 x: -41 y: -39 z: 40 ratio: -200 = -1 * 2^3 5^2
-182 x: -15 y: 71 z: 182 ratio: -182 = -1 * 2 7 13
-182 x: -85 y: -19 z: 182 ratio: -182 = -1 * 2 7 13
-168 x: -17 y: 26 z: 378 ratio: -168 = -1 * 2^3 3 7
-162 x: -1 y: 1 z: 18 ratio: -162 = -1 * 2 3^4
-162 x: -37 y: -35 z: 36 ratio: -162 = -1 * 2 3^4
-140 x: -37 y: -3 z: 140 ratio: -140 = -1 * 2^2 5 7
-128 x: -1 y: 1 z: 16 ratio: -128 = -1 * 2^7
-128 x: -33 y: -31 z: 32 ratio: -128 = -1 * 2^7
-127 x: -9 y: 44 z: 127 ratio: -127 = -127
-125 x: -4 y: 1 z: 35 ratio: -125 = -1 * 5^3
-125 x: -4 y: 7 z: 5 ratio: -125 = -1 * 5^3
-125 x: -7 y: -1 z: 20 ratio: -125 = -1 * 5^3
-121 x: -8 y: -3 z: 11 ratio: -121 = -1 * 11^2
-112 x: -159 y: 419 z: 532 ratio: -112 = -1 * 2^4 7
-104 x: -10 y: 17 z: 182 ratio: -104 = -1 * 2^3 13
-104 x: -25 y: -14 z: 26 ratio: -104 = -1 * 2^3 13
-104 x: -35 y: 3 z: 208 ratio: -104 = -1 * 2^3 13
-98 x: -1 y: 1 z: 14 ratio: -98 = -1 * 2 7^2
-98 x: -29 y: -27 z: 28 ratio: -98 = -1 * 2 7^2
-95 x: -219 y: -61 z: 380 ratio: -95 = -1 * 5 19
-81 x: -1 y: 4 z: 9 ratio: -81 = -1 * 3^4
-81 x: -8 y: 11 z: 117 ratio: -81 = -1 * 3^4
-78 x: -29 y: 53 z: 468 ratio: -78 = -1 * 2 3 13
-77 x: -4 y: 15 z: 77 ratio: -77 = -1 * 7 11
-74 x: -115 y: 3 z: 518 ratio: -74 = -1 * 2 37
-72 x: -16 y: 1 z: 78 ratio: -72 = -1 * 2^3 3^2
-72 x: -1 y: 1 z: 12 ratio: -72 = -1 * 2^3 3^2
-72 x: -1 y: 4 z: 18 ratio: -72 = -1 * 2^3 3^2
-72 x: -25 y: -23 z: 24 ratio: -72 = -1 * 2^3 3^2
-72 x: -29 y: -10 z: 42 ratio: -72 = -1 * 2^3 3^2
-72 x: -2 y: 5 z: 6 ratio: -72 = -1 * 2^3 3^2
-72 x: -46 y: 151 z: 258 ratio: -72 = -1 * 2^3 3^2
-56 x: -100 y: -41 z: 126 ratio: -56 = -1 * 2^3 7
-56 x: -17 y: 33 z: 28 ratio: -56 = -1 * 2^3 7
-56 x: -1 y: 2 z: 14 ratio: -56 = -1 * 2^3 7
-56 x: -1 y: 4 z: 14 ratio: -56 = -1 * 2^3 7
-56 x: -29 y: -19 z: 28 ratio: -56 = -1 * 2^3 7
-56 x: -40 y: 3 z: 182 ratio: -56 = -1 * 2^3 7
-56 x: -6 y: -1 z: 14 ratio: -56 = -1 * 2^3 7
-56 x: -73 y: -23 z: 112 ratio: -56 = -1 * 2^3 7
-50 x: -1 y: 1 z: 10 ratio: -50 = -1 * 2 5^2
-50 x: -21 y: -19 z: 20 ratio: -50 = -1 * 2 5^2
-49 x: -60 y: -19 z: 91 ratio: -49 = -1 * 7^2
-45 x: -11 y: -4 z: 15 ratio: -45 = -1 * 3^2 5
-43 x: -4 y: 15 z: 43 ratio: -43 = -43
-38 x: -13 y: 29 z: 152 ratio: -38 = -1 * 2 19
-35 x: -8 y: 23 z: 35 ratio: -35 = -1 * 5 7
-32 x: -17 y: -15 z: 16 ratio: -32 = -1 * 2^5
-32 x: -1 y: 1 z: 8 ratio: -32 = -1 * 2^5
-24 x: -2 y: 5 z: 18 ratio: -24 = -1 * 2^3 3
-23 x: -37 y: 60 z: 299 ratio: -23 = -23
-21 x: -11 y: 8 z: 63 ratio: -21 = -1 * 3 7
-18 x: -13 y: -11 z: 12 ratio: -18 = -1 * 2 3^2
-18 x: -1 y: 1 z: 6 ratio: -18 = -1 * 2 3^2
-17 x: -32 y: 83 z: 221 ratio: -17 = -17
-16 x: -15 y: 31 z: 104 ratio: -16 = -1 * 2^4
-16 x: -3 y: -1 z: 4 ratio: -16 = -1 * 2^4
-14 x: -5 y: 13 z: 28 ratio: -14 = -1 * 2 7
-13 x: -15 y: 32 z: 91 ratio: -13 = -13
-8 x: -134 y: -31 z: 190 ratio: -8 = -1 * 2^3
-8 x: -1 y: 1 z: 4 ratio: -8 = -1 * 2^3
-8 x: -1 y: 2 z: 2 ratio: -8 = -1 * 2^3
-8 x: -2 y: -1 z: 2 ratio: -8 = -1 * 2^3
-8 x: -4 y: 7 z: 18 ratio: -8 = -1 * 2^3
-8 x: -4 y: 9 z: 14 ratio: -8 = -1 * 2^3
-8 x: -95 y: 134 z: 62 ratio: -8 = -1 * 2^3
-8 x: -95 y: 31 z: 268 ratio: -8 = -1 * 2^3
-8 x: -9 y: -7 z: 8 ratio: -8 = -1 * 2^3
-7 x: -12 y: 17 z: 49 ratio: -7 = -7
-6 x: -17 y: -7 z: 18 ratio: -6 = -1 * 2 3
-5 x: -104 y: 89 z: 315 ratio: -5 = -5
-5 x: -4 y: -1 z: 5 ratio: -5 = -5
-2 x: -1 y: 1 z: 2 ratio: -2 = -2
-2 x: -295 y: 471 z: 392 ratio: -2 = -2
-2 x: -31 y: -1 z: 38 ratio: -2 = -2
-2 x: -5 y: -3 z: 4 ratio: -2 = -2
4 x: -1 y: -1 z: 2 ratio: 4 = 2^2
5 x: -3 y: -2 z: 5 ratio: 5 = 5
5 x: 4 y: 51 z: 95 ratio: 5 = 5
7 x: -3 y: -2 z: 7 ratio: 7 = 7
7 x: -44 y: 123 z: 133 ratio: 7 = 7
9 x: -137 y: -67 z: 234 ratio: 9 = 3^2
9 x: -2 y: -1 z: 3 ratio: 9 = 3^2
9 x: 4 y: 5 z: 21 ratio: 9 = 3^2
12 x: -121 y: -65 z: 342 ratio: 12 = 2^2 3
13 x: -11 y: -6 z: 13 ratio: 13 = 13
13 x: 1 y: 4 z: 13 ratio: 13 = 13
13 x: -245 y: -106 z: 403 ratio: 13 = 13
13 x: -249 y: -206 z: 247 ratio: 13 = 13
13 x: -7 y: -5 z: 26 ratio: 13 = 13
19 x: -18 y: -11 z: 19 ratio: 19 = 19
20 x: -7 y: -3 z: 10 ratio: 20 = 2^2 5
27 x: 19 y: 21 z: 156 ratio: 27 = 3^3
27 x: -19 y: 52 z: 63 ratio: 27 = 3^3
27 x: -21 y: 52 z: 57 ratio: 27 = 3^3
27 x: -2 y: -1 z: 9 ratio: 27 = 3^3
27 x: -3 y: -1 z: 6 ratio: 27 = 3^3
27 x: -3 y: -2 z: 3 ratio: 27 = 3^3
32 x: -1 y: 3 z: 4 ratio: 32 = 2^5
32 x: 3 y: 29 z: 112 ratio: 32 = 2^5
37 x: -38 y: -27 z: 37 ratio: 37 = 37
49 x: 19 y: 32 z: 259 ratio: 49 = 7^2
49 x: -1 y: 4 z: 7 ratio: 49 = 7^2
49 x: -23 y: -10 z: 133 ratio: 49 = 7^2
49 x: 24 y: 71 z: 455 ratio: 49 = 7^2
49 x: -29 y: 165 z: 364 ratio: 49 = 7^2
49 x: -3 y: -1 z: 14 ratio: 49 = 7^2
49 x: -51 y: -38 z: 49 ratio: 49 = 7^2
49 x: -5 y: -2 z: 7 ratio: 49 = 7^2
57 x: -343 y: -221 z: 342 ratio: 57 = 3 19
63 x: -22 y: -17 z: 21 ratio: 63 = 3^2 7
67 x: -17 y: 160 z: 469 ratio: 67 = 67
72 x: 1 y: 2 z: 18 ratio: 72 = 2^3 3^2
72 x: -89 y: 164 z: 126 ratio: 72 = 2^3 3^2
79 x: -83 y: -66 z: 79 ratio: 79 = 79
81 x: -22 y: -5 z: 63 ratio: 81 = 3^4
96 x: 1 y: 5 z: 36 ratio: 96 = 2^5 3
97 x: -102 y: -83 z: 97 ratio: 97 = 97
98 x: -1 y: 9 z: 28 ratio: 98 = 2 7^2
108 x: -5 y: -1 z: 18 ratio: 108 = 2^2 3^3
112 x: 1 y: 3 z: 28 ratio: 112 = 2^4 7
117 x: -41 y: -34 z: 39 ratio: 117 = 3^2 13
119 x: -11 y: -6 z: 119 ratio: 119 = 7 17
125 x: -18 y: -5 z: 35 ratio: 125 = 5^3
125 x: -18 y: -7 z: 25 ratio: 125 = 5^3
125 x: -7 y: -5 z: 90 ratio: 125 = 5^3
128 x: 1 y: 11 z: 72 ratio: 128 = 2^7
134 x: 7 y: 9 z: 134 ratio: 134 = 2 67
135 x: -298 y: -167 z: 315 ratio: 135 = 3^3 5
139 x: -146 y: -123 z: 139 ratio: 139 = 139
148 x: -7 y: -3 z: 74 ratio: 148 = 2^2 37
163 x: -171 y: -146 z: 163 ratio: 163 = 163
169 x: -7 y: -2 z: 13 ratio: 169 = 13^2
189 x: -22 y: -19 z: 21 ratio: 189 = 3^3 7
196 x: -5 y: -1 z: 14 ratio: 196 = 2^2 7^2
200 x: -16 y: 141 z: 490 ratio: 200 = 2^3 5^2
200 x: -3 y: 8 z: 10 ratio: 200 = 2^3 5^2
