Identifying relation between numbers based on equation relating them

Question

If $a^2 + b^2+16c^2=2(3ab+6bc + 4ac)$ , where $a,b,c$ are non zero numbers. Then $a,b,c$ are in __________?

1. Harmonic progression 2. Geometric progression 3. Arithmetic progression 4. None of these

My attempt:

$ a \rightarrow 4a'$

$ b \rightarrow 4b'$

$a'^2 +b'^2 +c^2 - 6a'b' -3b'c -2a'c=0$

Ok, this looks nicer than the original thing but still the solution doesn't seem in sight. Further, I started wondering, would the relationship between numbers be preserved under transformations to the equation?

Answer 1

Try $a=b=4$ and $c=\frac{5+\sqrt{41}}{2}.$

We got: non one of them.

Answer 2

Hint:

$$a^2-2a(3b+4c)+b^2+16c^2-12bc=0$$

$$\implies a=\dfrac{2(3b+4c)\pm\sqrt{32b^2+96bc}}2$$

For values of $a,$ we just need $b(b+3c)\ge0$

In that at least one of the two values of $a$ will be $\ne0$

Answer 3

There is a way to make this appear simpler. You are using letters a,b,c. Alright, take any $x,y,z$ you like such that $2x^2 - 16 y^2 + 81 z^2 = 0.$ This is just a point on a cone. Then still to avoid fractions, let $$ a = 4x+12y - 11z \; , \; \; b = 4y - 9 z \; , \; \; c = 4z $$

For infinitely many examples, we can introduce variables $u,v$ and take $$ x = 576 u^2 + 72uv \; , \; \; y = 594 u^2 + 144 uv + 9 v^2 \; , \; \; z = 248 u^2 + 64 uv + 4 v^2 $$

For instance, let $u=1, v=1$ to get $x=648,y=747, z=316,$ then $a=8080, b= 144, c = 1264.$ These are all multiples of $16,$ we can divide out to get $a = 505, b= 9, c=79$

Here is a good one, as a single step: $$ a = 16u^2 + 36 uv + 19 v^2 \; , \; \; b = 9 v^2 \; , \; \; c = 4 u^2 - 2 v^2 $$ With $u=0, v=1$ we get $a=19,b=9, c=-2,$ or with $u=1,v=1$ we get $a= 71, b=9,c=2$

$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 3 & 1 & 0 \\ - \frac{ 11 }{ 4 } & - \frac{ 9 }{ 4 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - 3 & - 4 \\ - 3 & 1 & - 6 \\ - 4 & - 6 & 16 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 3 & - \frac{ 11 }{ 4 } \\ 0 & 1 & - \frac{ 9 }{ 4 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 8 & 0 \\ 0 & 0 & \frac{ 81 }{ 2 } \\ \end{array} \right) $$

Answer 4

I worked it all out. I like to use $x,y,z$ for the variables, so we are asking about integer triples with $x > 0$ and $\gcd(x,y,z) = 1$ so that $$ x^2 + y^2 + 16 z^2 - 12yz-8zx-6xy = 0. $$ Oh, there cannot be any integer solutions with $x=0$ as $\sqrt 5$ is irrational.

There is a theorem in Fricke and Klein (1897) that says that this problem can be parametrized by a finite number of recipes of the Pythagorean type. In this case, three of them.

A: $$x = u^2 - 12uv + 16v^2$$ $$y = 9u^2$$ $$z = 7u^2 - 12 uv + 4v^2$$

B $$x = u^2 +4uv - 16v^2$$ $$y = -9u^2$$ $$z = -2u^2 - 8 uv - 4v^2$$

C $$x = 4u^2 +4uv - 4v^2$$ $$y = 36v^2$$ $$z = u^2 - 8 uv +8v^2$$

here is a list with $x$ up to 1000 while $\gcd(x,y,z) = 1.$ I did a brute force search up to 1000. Then, for each recipe above, I printed out the triple with the name ABC and, if $x$ turned out negative, simply negated all three and printed $-x,-y,-z.$

Thu Aug  6 17:38:05 PDT 2020
Thu Aug  6 17:38:59 PDT 2020
         1      -225      -146
         1      -225      -146   B  5         2
         1        81         7
         1        81         7   A  3         2
         1        -9        -2
         1        -9        -2   B  1         0
         1         9         7
         1         9         7   A  1         0
         4         0         1
         4         0         1   C  1         0
         4      -144       -17
         4      -144       -17   C  1         2
         4      -144       -89
         4      -144       -89   C  3        -2
         4       324       217
         4       324       217   C  5        -3
         4        36         1
         4       -36       -17
         4       -36       -17   C  1        -1
         4        36         1   C  1         1
         4      -900       -89
         4      -900       -89   C  3         5
         5      -441       -46
         5      -441       -46   B  7        -1
         5       -81       -46
         5       -81       -46   B  3         1
         5         9        -1
         5         9        -1   A  1         1
        11       225        14
        11       225        14   B  5        -1
        11      -225       -31
        11      -225       -31   A  5         3
        11       441       302
        11       441       302   B  7         3
        11      -729      -463
        11      -729      -463   A  9         1
        11       -81       -31
        11       -81       -31   A  3         1
        11         9        14
        11         9        14   B  1         1
        19      -225      -119
        19      -225      -119   A  5         1
        19      -441      -263
        19      -441      -263   A  7         1
        19        81        -2
        19        81        -2   B  3        -1
        19         9        -2
        19         9        -2   B  1        -1
        20      -144       -49
        20      -144       -49   C  1        -2
        20      -324       -49
        20      -324       -49   C  1         3
        20        36        41
        20        36        41   C  3        -1
        20       576       401
        20       576       401   C  7        -4
        20       576        41
        20       576        41   C  3         4
        29      -225       -94
        29      -225       -94   B  5         1
        29       441        23
        29       441        23   A  7         5
        29      -729       -94
        29      -729       -94   B  9        -1
        29         9        23
        29         9        23   A  1        -1
        31      -225       -71
        31      -225       -71   A  5         2
        31      -441       -71
        31      -441       -71   A  7         4
        31       729       514
        31       729       514   B  9         4
        31        81        82
        31        81        82   B  3         2
        41       225        -1
        41       225        -1   A  5         4
        41      -441      -226
        41      -441      -226   B  7         2
        41         9        -1
        41         9        -1   A  1         2
        44       144       137
        44       144       137   C  5        -2
        44       144        -7
        44       144        -7   C  3         2
        44      -324       -97
        44      -324       -97   C  1        -3
        44        36        -7
        44        36        -7   C  3         1
        44      -576      -313
        44      -576      -313   C  5        -4
        44      -576       -97
        44      -576       -97   C  1         4
        44      -900      -529
        44      -900      -529   C  7        -5
        44       900       641
        44       900       641   C  9        -5
        55      -441      -191
        55      -441      -191   A  7         2
        55       729        34
        55       729        34   B  9        -2
        55         9        34
        55         9        34   B  1         2
        59       225       206
        59       225       206   B  5         3
        59      -441      -127
        59      -441      -127   A  7         3
        59      -729      -127
        59      -729      -127   A  9         5
        61      -441      -158
        61      -441      -158   B  7         1
        61        81       103
        61        81       103   A  3        -1
        71       441         2
        71       441         2   B  7        -2
        71      -729      -367
        71      -729      -367   A  9         2
        71         9         2
        71         9         2   B  1        -2
        76       324       289
        76       324       289   C  7        -3
        76        36        73
        76        36        73   C  5        -1
        76      -576      -161
        76      -576      -161   C  1        -4
        76      -576      -233
        76      -576      -233   C  3        -4
        76      -900      -161
        76      -900      -161   C  1         5
        79       225       -14
        79       225       -14   B  5        -2
        79        81       -14
        79        81       -14   B  3        -2
        89      -729      -322
        89      -729      -322   B  9         2
        89         9        47
        89         9        47   A  1        -2
        95       441       386
        95       441       386   B  7         4
        95      -729      -199
        95      -729      -199   A  9         4
       101       225       239
       101       225       239   A  5        -1
       101      -729      -238
       101      -729      -238   B  9         1
       109       729         7
       109       729         7   A  9         7
       109         9         7
       109         9         7   A  1         3
       116        36        -7
       116        36        -7   C  5         1
       116       576       497
       116       576       497   C  9        -4
       116       576        -7
       116       576        -7   C  5         4
       116      -900      -241
       116      -900      -241   C  1        -5
       121       441       -17
       121       441       -17   A  7         6
       121        81       -17
       121        81       -17   A  3         4
       124       144       193
       124       144       193   C  7        -2
       124       144       -23
       124       144       -23   C  5         2
       124       324       -23
       124       324       -23   C  5         3
       124      -900      -329
       124      -900      -329   C  3        -5
       131         9        62
       131         9        62   B  1         3
       139       729       622
       139       729       622   B  9         5
       145        81       151
       145        81       151   A  3        -2
       149       441       431
       149       441       431   A  7        -1
       151       225       274
       151       225       274   B  5         4
       155         9        14
       155         9        14   B  1        -3
       164        36       113
       164        36       113   C  7        -1
       164       900       761
       164       900       761   C  11        -5
       179       225       -34
       179       225       -34   B  5        -3
       179       441       -34
       179       441       -34   B  7        -3
       181         9        79
       181         9        79   A  1        -3
       199        81       178
       199        81       178   B  3         4
       205       729       679
       205       729       679   A  9        -1
       209       225       311
       209       225       311   A  5        -2
       209         9        23
       209         9        23   A  1         4
       211       441       478
       211       441       478   B  7         5
       220        36         1
       220        36         1   C  7         1
       229        81       -17
       229        81       -17   A  3         5
       236       144       257
       236       144       257   C  9        -2
       236       144       -31
       236       144       -31   C  7         2
       236       900       -31
       236       900       -31   C  7         5
       239         9        98
       239         9        98   B  1         4
       241       225       -41
       241       225       -41   A  5         6
       241       729       -41
       241       729       -41   A  9         8
       244       324       -47
       244       324       -47   C  7         3
       244       576       -47
       244       576       -47   C  7         4
       244       576       601
       244       576       601   C  11        -4
       271         9        34
       271         9        34   B  1        -4
       281       441       527
       281       441       527   A  7        -2
       284        36       161
       284        36       161   C  9        -1
       295        81       -14
       295        81       -14   B  3        -4
       305         9       119
       305         9       119   A  1        -4
       311       225       -46
       311       225       -46   B  5        -4
       316       324       457
       316       324       457   C  11        -3
       316       900       889
       316       900       889   C  13        -5
       319       441       -62
       319       441       -62   B  7        -4
       319       729       -62
       319       729       -62   B  9        -4
       331        81       238
       331        81       238   B  3         5
       341         9        47
       341         9        47   A  1         5
       349       225       391
       349       225       391   A  5        -3
       356        36        17
       356        36        17   C  9         1
       359       441       578
       359       441       578   B  7         6
       361       729       799
       361       729       799   A  9        -2
       379         9       142
       379         9       142   B  1         5
       380       144       -31
       380       144       -31   C  9         2
       380       144       329
       380       144       329   C  11        -2
       389       225       -49
       389       225       -49   A  5         7
       401       441       -73
       401       441       -73   A  7         8
       404       576       713
       404       576       713   C  13        -4
       404       576       -79
       404       576       -79   C  9         4
       404       900       -79
       404       900       -79   C  9         5
       409        81       271
       409        81       271   A  3        -4
       419         9        62
       419         9        62   B  1        -5
       431       225       434
       431       225       434   B  5         6
       436        36       217
       436        36       217   C  11        -1
       445       441       631
       445       441       631   A  7        -3
       451       729       862
       451       729       862   B  9         7
       451        81        -2
       451        81        -2   B  3        -5
       461         9       167
       461         9       167   A  1        -5
       484       324       553
       484       324       553   C  13        -3
       491       441       -82
       491       441       -82   B  7        -5
       499       729       -98
       499       729       -98   B  9        -5
       505         9        79
       505         9        79   A  1         6
       521       225       479
       521       225       479   A  5        -4
       524        36        41
       524        36        41   C  11         1
       541        81         7
       541        81         7   A  3         7
       551         9       194
       551         9       194   B  1         6
       556       144       -23
       556       144       -23   C  11         2
       556       144       409
       556       144       409   C  13        -2
       569       225       -49
       569       225       -49   A  5         8
       580       324       -71
       580       324       -71   C  11         3
       589       441       -89
       589       441       -89   A  7         9
       589        81       343
       589        81       343   A  3        -5
       596       576      -103
       596       576      -103   C  11         4
       596       576       833
       596       576       833   C  15        -4
       599         9        98
       599         9        98   B  1        -6
       601       729      -113
       601       729      -113   A  9        10
       604       900      -119
       604       900      -119   C  11         5
       619       225       526
       619       225       526   B  5         7
       620        36       281
       620        36       281   C  13        -1
       641       441       743
       641       441       743   A  7        -4
       649         9       223
       649         9       223   A  1        -6
       655       729       994
       655       729       994   B  9         8
       671       225       -46
       671       225       -46   B  5        -6
       691        81       382
       691        81       382   B  3         7
       695       441       -94
       695       441       -94   B  7        -6
       701         9       119
       701         9       119   A  1         7
       724        36        73
       724        36        73   C  13         1
       745        81        31
       745        81        31   A  3         8
       751       441       802
       751       441       802   B  7         8
       755         9       254
       755         9       254   B  1         7
       764       144       497
       764       144       497   C  15        -2
       764       144        -7
       764       144        -7   C  13         2
       781       225       -41
       781       225       -41   A  5         9
       796       324       -71
       796       324       -71   C  13         3
       809       441       -97
       809       441       -97   A  7        10
       811         9       142
       811         9       142   B  1        -7
       820       576      -119
       820       576      -119   C  13         4
       820       576       961
       820       576       961   C  17        -4
       829       729      -137
       829       729      -137   A  9        11
       836        36       353
       836        36       353   C  15        -1
       836       900      -151
       836       900      -151   C  13         5
       839       225       626
       839       225       626   B  5         8
       859        81        46
       859        81        46   B  3        -7
       869       441       863
       869       441       863   A  7        -5
       869         9       287
       869         9       287   A  1        -7
       899       225       -34
       899       225       -34   B  5        -7
       916       324       769
       916       324       769   C  17        -3
       919        81       466
       919        81       466   B  3         8
       929         9       167
       929         9       167   A  1         8
       955       729      -146
       955       729      -146   B  9        -7
       956        36       113
       956        36       113   C  15         1
       961       225       679
       961       225       679   A  5        -6
       991         9       322
       991         9       322   B  1         8
       995       441       926
       995       441       926   B  7         9