Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$

Question

I'm something like 90% sure that this diophantine equation has nontrivial solutions:

$3(x^2+y^2+z^2)=10(xy+yz+zx)$

However, I have not been able to find a solution using my calculator. I would greatly appreciate if someone could try to find one using a program. Or maybe you can just guess one that happens to work?

Thanks!

EDIT: By nontrivial I mean no $0$'s. (Credits to Slade for reminding me to define this)

EDIT2: In fact, you are free to find a nontrivial solution to $(3n-3)(x^2+y^2+z^2)=(9n+1)(xy+yz+zx)$ where $n\equiv 1\pmod 5$ is a positive integer. The one I posted above is the case $n=5(2)+1$, but you will make my day if you can find a nontrivial solution for any $n=5k+1$.

Answer 1

As far as I understand - this is the site for solving the problem. Programming and calculation using the computer is not mathematics. If you want to calculate - there is a special section. https://mathematica.stackexchange.com/questions

Here it is necessary to solve the equations.

For the equation:

$$3(x^2+y^2+z^2)=10(xy+xz+yz)$$

The solution is simple.

$$x=4ps$$

$$y=3p^2-10ps+7s^2$$

$$z=p^2-10ps+21s^2$$

$p,s - $ any integer which we ask.

Why make a program? What's the point? For what?

Answer 2

When he wrote the equation he meant probably that entry.

$$q(x^2+y^2+z^2)=(3q+1)(xy+xz+yz)$$

It turns out, this equation has a connection with the Pell equation:

$$p^2-5s^2=\pm1$$

For $+1$ it is necessary to use the first solution $(9 ; 4)$. For $-1$ it is necessary to use the first solution $(2 ; 1)$. Knowing what the decision can be found on the following formula.

$$p_2=9p_1+20s_1$$

$$s_2=4p_1+9s_1$$

Using the solutions of the Pell equation can be found when there are solutions. $q=\mp(p^2-s^2)$

Will make a replacement. $t=\mp4ps$ Then the solution can be written:

$$x=2(q+1)tkn$$

$$y=(q+t+1)k^2-2(3q+1)tkn+(t-q-1)(10q^2+7q+1)n^2$$

$$z=(t-q-1)k^2-2(3q+1)tkn+(t+q+1)(10q^2+7q+1)n^2$$

$k,n $ - integers asked us. May be necessary, after all the calculations is to obtain a relatively simple solution, divided by the common divisor.

Answer 3

This was a bunch of nonsense characters typed by hand so that the software would not test me with a ``captcha''

   0           1           3
   0           3           1
   1           0           3
   1           3           0
   3           0           1
   3           1           0
   3           9          40
   3          40           9
   5          32         119
   5         119          32
   8          11          65
   8          65          11
   9           3          40
   9          40           3
  11           8          65
  11          65           8
  13          15          96
  13          96          15
  15          13          96
  15          96          13
  32           5         119
  32         119           5
  40           3           9
  40           9           3
  65           8          11
  65          11           8
  96          13          15
  96          15          13
 119           5          32
 119          32           5

Answer 4

Because the equation is homogenous, the integer solutions can be derived from the rational solutions, in other words swapping between projective and affine form.

I prove below that the set of non-zero rational solutions are common rational multiples of the following (which, conversely, satisfies the equation identically) for any rational parameter t:

$x,\ y,\ z = 2 t - 1,\ 3 t^2 - 8 t + 5,\ t^2 - 6 t + 8$

So, explicitly, the complete set of integer solutions with GCD(x, y, z) = 1 can be expressed as follows, as $m,\ n$ range over coprime integer pairs

$x,\ y,\ z = (2 m - n) n,\ 3 m^2 - 8 m n + 5 n^2,\ m^2 - 6 m n + 8 n^2$

Proof:

Let $p,\ q,\ r = - x + y + z,\ x - y + z,\ x + y - z$

<=> $2 x,\ 2 y,\ 2 z = q + r,\ r + p,\ p + q$

Then $p^2 + q^2 + r^2 = 3 (x^2 + y^2 + z^2) - 2 (x y + y z + z x)$

and $p q + q r + r p = - (x^2 + y^2 + z^2) + 2 (x y + y z + z x)$

So if $a (p^2 + q^2 + r^2) = b (p q + q r + r p)$

then $(3 a + b) (x^2 + y^2 + z^2) = 2 (a + b) (x y + y z + z x)$

So the required equation is obtained with $3 a + b,\ a + b = 3,\ 5$, i.e. $a,\ b = -1,\ 6$ and the original is equivalent to

$p^2 + q^2 + r^2 + 6 (p q + q r + r p) = 0$

If r = 0 then this becomes $(p + 3 q)^2 = 8 q^2$, which for rational $p, q$ has only the solution p = q = 0

Otherwise, we can replace $\frac{p}{r},\ \frac{q}{r}$ by $p,\ q$ respectively and the equation becomes

$q^2 + 6 (p + 1) q + (p^2 + 6 p + 1) = 0$

which for rational $q$ (assuming rational $p$) requires rational $s$ with

$q = 2 s - 3 (p + 1)$

$9 (p + 1)^2 - (p^2 + 6 p + 1) = 4 s^2$

The latter is equivalent to $8 s^2 - (4 p + 3)^2 = 7$

So in view of the obvious rational solution $4 p + 3,\ s = 1, 1$ we can replace in this $4 p + 3,\ s = 2 t u + 1,\ u + 1$

which implies either $u = 0$, which recovers the solution already observed, or $u = \frac{4 - t}{t^2 - 2}$

which gives successively

$p = \frac{- t^2 + 2 t + 1}{t^2 - 2}$

$s = \frac{t^2 - t + 2}{t^2 - 2}$

$q = \frac{2 v^2 - 8 v + 7}{t^2 - 2}$

So the original $p,\ q,\ r$ can be taken as follows, and the result follows

$p,\ q,\ r = - t^2 + 2 t + 1,\ 2 t^2 - 8 t + 7,\ t^2 - 2$

Sanity check: In the expressions for $x, y, z$ take $t = 0$ to give $x, y, z = 1, 5, 8$ and the equation becomes 2.3^3.5 = 2.3^3.5

Regards

John R Ramsden

Answer 5

august 2020, this is the good answer, gradually deleting my others here, limit of 5 per day

ADDED: Another way of saying this: Given integers $B > A > 0,$ with $\gcd(A,B) = 1,$ then there is a solution in integers $x,y,z,$ not all $0,$ to $$ A(x^2 + y^2 + z^2) - B (yz + zx + xy) = 0, $$ if and only if both $B+2A$ and $B-A$ are integrally represented by the binary form $u^2 + 3 v^2.$

ORIGINAL: It is simpler than I had feared. We take $0 < A < B,$ and $\gcd(A,B)=1.$ After that, what we are really concerned about are the two numbers that come up in diagonalizing the form, those being $B + 2A$ and $B-A.$

The form is isotropic over the rationals (and integers) if and only if:

(I) when factoring both $B + 2A$ and $B-A,$ the exponents of $2$ are even.

(II) when factoring both $B + 2A$ and $B-A,$ the exponents of $q$ are even, where $q \equiv 5 \pmod 6$ is a prime.

That's it. Note that we could combine these into one test, for all primes $p \equiv 2 \pmod 3.$

Here is an example, $$ 6(x^2 + y^2 + z^2) = 55 (yz+zx+xy) $$ All primitive solutions can be found by ordering the elements of three Pythagorean Triple type recipes for $(x,y,z).$ The theorem is that a finite number of such recipes succeed; for this problem the count required has turned out to be one of $1,2,3,4,6,8,12,16.$ That is, either $2^k$ or $3 \cdot 2^k$ The quadratic form is so symmetric that each matrix of coefficients comes out in an amusing cyclic pattern.

$$ x= 48 u^2 + 97uv + 34v^2 \; , \; \; y= 34 u^2 -29uv -15v^2 \; , \; \; z = -15 u^2 -uv + 48 v^2 $$

$$ x= 54 u^2 + 91uv + 25v^2 \; , \; \; y= 25 u^2 -41uv -12v^2 \; , \; \; z = -12 u^2 +17uv + 54 v^2 $$

$$ x= 60 u^2 + 71uv + 9v^2 \; , \; \; y= 9 u^2 -53uv -2v^2 \; , \; \; z = -2 u^2 +49uv + 60 v^2 $$

Here are the three recipes sorted and with the $u,v$ values specified.

august 2020 my good answer

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
         x         y         z  such that x >= |y| >= |z| 
        48        34       -15
        48        34       -15      < 48, 97, 34 >      1  0    
        54        25       -12
        54        25       -12      < 54, 91, 25 >      1  0    
        60         9        -2
        60         9        -2      < 60, 71, 9 >      1  0    
       140       107       -46
       140       107       -46      < 60, 71, 9 >      1  1    
       170        59       -28
       170        59       -28      < 54, 91, 25 >      1  1    
       179        32       -10
       179        32       -10      < 48, 97, 34 >      1  1    
       391       150       -72
       391       150       -72      < 60, 71, 9 >      2  1    
       423        40         6
       423        40         6      < 54, 91, 25 >      2  1     POSITIVE 
       552       525      -206
       552       525      -206      < 54, 91, 25 >      1  3    
       645       414      -188
       645       414      -188      < 48, 97, 34 >      1  3    
       685       354      -168
       685       354      -168      < 60, 71, 9 >      1  3    
       757       204       -90
       757       204       -90      < 48, 97, 34 >      3  1    
       762       189       -80
       762       189       -80      < 60, 71, 9 >      3  1    
       784        90        -3
       784        90        -3      < 54, 91, 25 >      3  1    
       826       747      -300
       826       747      -300      < 60, 71, 9 >      2  3    
       920       818      -331
       920       818      -331      < 54, 91, 25 >      1  4    
       987       540      -254
       987       540      -254      < 54, 91, 25 >      2  3    
      1002       516      -245
      1002       516      -245      < 60, 71, 9 >      3  2    
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
         x         y         z  such that x >= |y| >= |z|

august 2020 my good answer

Answer 6

$(3n-3)(x^2+y^2+z^2)=(9n+1)(xy+yz+zx)\implies$

$(7 + 3 n) \Biggl(6 (n - 1) x - (1 + 9 n) (y + z)\Biggr)^2 =\\ 5 (-1 + 3 n) \Bigl(\Biggl(y (7 + 3 n) + z (1 + 9 n)\Biggr)^2 - 24 (-1 + n) (2 + 3 n) z^2\Bigl)$

Let $n,z$ as parameters, then this equation have Pell form.

gp-code:

nxyz()=
{
 for(n=2, 55,
  n19= 1+9*n; n73= 7+3*n;
  D= 5*(3*n-1)*n73;
  if(!issquare(D),
   C= -5*(3*n-1)*24*(n-1)*(2+3*n)*n73;
   Q= bnfinit('x^2-D, 1);
   fu= Q.fu[1]; \\print("Fundamental Unit: "fu);
   N= bnfisintnorm(Q, C); \\print("Fundamental Solutions (Norm): "N"\n");
   for(i=1, #N, ni= N[i];
    for(j=0, 3,
     s= lift(ni*fu^j);
     X= abs(polcoeff(s, 0)); Y= abs(polcoeff(s, 1));  
     if(X^2-D*Y^2==C,
      y= (Y-n19)/n73;
      x= (X/n73+n19*(y+1))/6/(n-1);
      z= lcm(denominator(x),denominator(y));
      x= x*z; y= y*z;
      if(3*(n - 1)*(x^2 + y^2 + z^2) == (9*n + 1)*(x*y + y*z + z*x),
       print("("n", "x", "y", "z")")
      )
     )
    )
   )
  )
 )
};

Output $(n,x,y,z)$:

? \r nxyz.gp
? nxyz()
(2, 263, 3, 39)
(2, 263, 39, 3)
(2, 13, 1, 1)
(2, 2029, 313, 13)
(12, 31259, 9501, 645)
(12, 17390279, 5863461, 15)
(12, 17008338675311, 5734707254439, 645)
(12, 8996620057194221, 3033393402029559, 15)
(12, 23, -3, 15)
(12, 54305687, 18309183, 645)
(12, 28726047317, 9685570983, 15)
(12, 28095079670561213, 9472827435234027, 645)
(12, 27, 3, 5)
(12, 248102969, 83641041, 7095)
(12, 131244443339, 44251732701, 165)
(12, 128361659301988751, 43279743717614349, 7095)
(12, 22431, 7209, 215)
(12, 418643, 140877, 165)
(12, 409856470367, 138191431653, 7095)
(12, 216795018826247, 73096849207503, 165)
(12, 335, -9, 129)
(12, 331085, 111627, 3)
(12, 323820154601, 109182549915, 129)
(12, 171285800223257, 57752490708225, 3)
(12, 345, 99, 11)
(12, 1033625, 348291, 129)
(12, 546912497, 184402665, 3)
(12, 534899570406731, 180352267551945, 129)
(12, 79, 25, 1)
(12, 4720355, 1589181, 1419)
(12, 2498750495, 842504487, 33)
(12, 2443865517463541, 823998956160435, 1419)
(12, 48157, 16165, 43)
(12, 7895, 2607, 33)
(12, 7803215405, 2631010371, 1419)
(12, 4127539903577, 1391684013555, 33)
(12, 157553, 53097, 15)
(12, 154129532603, 51967905117, 645)
(12, 81527354638727, 27488605506693, 15)
(12, 79736606139995013647, 26884818234807767943, 645)
(12, 490499, 164301, 645)
(12, 260315351, 87770589, 15)
(12, 254597561009591, 85842745033119, 645)
(12, 134670267767599349, 45406819349798631, 15)
(12, 2230517, 740223, 7095)
(12, 1189336493, 401008707, 165)
(12, 1163213123733593, 392200959043377, 7095)
(12, 615285638339258987, 207456065037378333, 165)
(12, 3389, 891, 165)
(12, 3714100769, 1252272681, 7095)
(12, 1964596066931, 662403514509, 165)
(12, 1921443711015792671, 647853820469569989, 7095)
(12, 2534577, 854223, 215)
(12, 1340961597, 452132463, 5)
(12, 1311507385382253, 442201384990917, 215)
(12, 693726405183032343, 233903964702072237, 5)
(12, 4269, 1431, 5)
(12, 4187619141, 1411940589, 215)
(12, 2215056036591, 746851186749, 5)
(12, 2166402326572054479, 730446599032512561, 215)
(12, 19423, 6457, 55)
(12, 19132518907, 6450913843, 2365)
(12, 10120215768577, 3412236544783, 55)
(12, 9897925210224031573, 3337286762762271847, 2365)
(12, 55219, 14971, 2365)
(12, 32313529, 10895071, 55)
(12, 31603898624581, 10655897094799, 2365)
(12, 16716992394487381, 5636473932303709, 55)
(12, 48157, 16165, 43)
(12, 25530439, 8608105, 1)
(12, 24969665338195, 8419030436497, 43)
(12, 13207791558649195, 4453275509545459, 1)
(12, 79, 25, 1)
(12, 79727587, 26881705, 43)
(12, 42172242835, 14219229259, 1)
(12, 41245929443880625, 13906903863923347, 43)
(12, 345, 99, 11)
(12, 364261137, 122817255, 473)
(12, 192677832909, 64965249525, 11)
(12, 188445663972161535, 63538287723634977, 473)
(12, 335, -9, 129)
(12, 615189, 207405, 11)
(12, 601703642247, 202876618065, 473)
(12, 318273240551715, 107312295261369, 11)
(12, 22431, 7209, 215)
(12, 12151779, 4097211, 5)
(12, 11884877546769, 4007228148111, 215)
(12, 6286547425375101, 2119637303837199, 5)
(12, 27, 3, 5)
(12, 37947657, 12794463, 215)
(12, 20072833773, 6767964087, 5)
(12, 19631933953623333, 6619305755429517, 215)
(12, 23, -3, 15)
(12, 173372911, 58452229, 2365)
(12, 91709376799, 30921681241, 55)
(12, 89694980300581339, 30242486590433491, 2365)
(12, 31259, 9501, 645)
(12, 292687, 98593, 55)
(12, 286394466067, 96563714203, 2365)
(12, 151489354754713, 51077716550047, 55)
(21, 1417, 426, 48)
(21, 83142331513, 29578439082, 336)
(21, 91315532516753401, 32486110508186634, 48)
(21, 4914317238221204673953257, 1748301175858017549085194, 336)
(21, 2954, 1037, 8)
(21, 87347683, 31073934, 336)
(21, 95935286323747, 34129618776174, 48)
(21, 5162937983728230318259, 1836749666389321064622, 336)
(21, 12839986, 4567617, 168)
(21, 20199, 7158, 16)
(21, 1089383306103, 387555385974, 112)
(21, 1196473731707640183, 425653520240554902, 16)
(21, 166, 27, 24)
(21, 1144496109, 407161938, 112)
(21, 1257004661384493, 447187802705394, 16)
(21, 67648071535180409104413, 24066253210037399689266, 112)
(21, 162, 51, 4)
(21, 376240009, 133849602, 168)
(21, 413226300173881, 147008015858322, 24)
(21, 22238551035757462803577, 7911513042504815726946, 168)
(21, 164382, 58473, 4)
(21, 394819, 140166, 168)
(21, 434131875763, 154445313942, 24)
(21, 23363623933881420691, 8311765239375213222, 168)
(21, 71, 14, 8)
(21, 4929739847, 1753787774, 56)
(21, 5414351751364439, 1926191793195182, 8)
(21, 291383529309669591514199, 103661636443752927469598, 56)
(21, 12514, 4431, 12)
(21, 5178989, 1842362, 56)
(21, 5688269796029, 2023639966826, 8)
(21, 306124944398220177437, 108905993306325509018, 56)
(21, 9536, 3389, 2)
(21, 1702357, 605478, 84)
(21, 1869955422109, 665249129166, 12)
(21, 100635170298651434053, 35801633886681667206, 84)
(21, 1567, 426, 84)
(21, 1964558503, 698904786, 12)
(21, 105726414890938351, 37612878150289098, 84)
(21, 116119714172828328814711, 41310363777438054629874, 12)
(21, 712, 243, 6)
(21, 22308267, 7936266, 28)
(21, 24501335976387, 8716513896258, 4)
(21, 1318585507084135954491, 469095599806853388714, 28)
(21, 23361, 8262, 28)
(21, 25740885657, 9157491966, 4)
(21, 1385294206562535537, 492827665134459174, 28)
(21, 1521473772446088640861641, 541274455119923650488798, 4)
(21, 7591, 2628, 42)
(21, 8462029819, 3010423608, 6)
(21, 455400061126042903, 162011612958193764, 42)
(21, 500167578620665908843883, 177937956269400884789592, 6)
(21, 438, 144, 7)
(21, 8890117, 3162708, 6)
(21, 478439254001089, 170207959681728, 42)
(21, 525471609729598654933, 186940034319557009844, 6)
(21, 40222, 14304, 3)
(21, 100913, 35876, 14)
(21, 110874854861, 39444470048, 2)
(21, 5966938981113611201, 2122778390417863172, 14)
(21, 71, 8, 14)
(21, 116484131, 41440004, 2)
(21, 6268813025048183, 2230172097496232, 14)
(21, 6885060629573009104979, 2449406298176474029988, 2)
(21, 71, 8, 14)
(21, 38292856, 13622934, 3)
(21, 2060802551018062, 733144269794844, 21)
(21, 2263387096192637014984, 805215074623013954982, 3)
(21, 100913, 35876, 14)
(21, 40222, 14304, 3)
(21, 2165060831764, 770234849382, 21)
(21, 2377894353243482254, 845951796011137296, 3)
(21, 438, 144, 7)
(21, 501737268, 178496382, 1)
(21, 27001935493032294, 9606118873595136, 7)
(21, 29656326044900147464932, 10550436038739482164974, 1)
(21, 7591, 2628, 42)
(21, 527118, 187524, 1)
(21, 28367993280288, 10092103054086, 7)
(21, 31156672386592834974, 11084194269277534932, 1)
(21, 23361, 8262, 28)
(21, 693124, 246573, 6)
(21, 37302648961168, 13270666479129, 42)
(21, 40969637906871469348, 14575222285199263197, 6)
(21, 712, 243, 6)
(21, 39189831124, 13942044063, 42)
(21, 43042337210924968, 15312598904334339, 6)
(21, 2316404382684519800324788, 824076328347877344913071, 42)
(21, 1567, 426, 84)
(21, 9081956, 3230963, 2)
(21, 488762846529896, 173880646666007, 14)
(21, 536810049748675969988, 190973760075715894979, 2)
(21, 9536, 3389, 2)
(21, 513489899588, 182677460897, 14)
(21, 563967864007424864, 200635333861190573, 2)
(21, 30350991989079117832003172, 10797568087438115061341201, 14)
(21, 12514, 4431, 12)
(21, 675216370618, 240212732871, 84)
(21, 741592648046669506, 263826536981350143, 12)
(21, 39910204031994469833921946, 14198321609193718386673527, 84)
(21, 71, 14, 8)
(21, 709376086, 252365133, 84)
(21, 779110640492446, 277173813343269, 12)
(21, 41929305404327431789558, 14916630406186233794397, 84)
(21, 394819, 140166, 168)
(21, 164382, 58473, 4)
(21, 8847110986086, 3147418815249, 28)
(21, 9716814856799508798, 3456821775284131641, 4)
(21, 162, 51, 4)
(21, 9294696042, 3306650139, 28)
(21, 10208399269556226, 3631706212978083, 4)
(21, 549384218898584896608714, 195447104722505639044491, 28)
(21, 166, 27, 24)
(21, 12222111382, 4348097451, 168)
(21, 13423590828090502, 4775532081335883, 24)
(21, 722415803611620115127926, 257004246508466775463899, 168)
(21, 20199, 7158, 16)
(21, 12839986, 4567617, 168)
(21, 14102705137762, 5017131532161, 24)
(21, 758963618283224242642, 270006375648288958929, 168)
(21, 2954, 1037, 8)
(21, 160141821914, 56971522493, 56)
(21, 175884357987566762, 62572034929515101, 8)
(21, 9465547739462478924449018, 3367431820301536704807437, 56)
(21, 1417, 426, 48)
(21, 168243422, 59853623, 56)
(21, 184782542228846, 65737623396407, 8)
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