Find positive integer values of $x,y,z,w$, such that $$xyzw=504(x^2+y^2+z^2+w^2)$$
I found it at some point and now I am unable to find the solution anymore, maybe this equation isn't satisfiable? But how do one prove such thing?
Edit @Dashisan found nice example $(x,y,z,w)=(21,63,84,84)$. Now, the problem is if there exists distinct positive integer solution.
On
Tell you what. It happens that $$ 2xyzw = x^2 + y^2 + z^2 + w^2 $$ is not possible in positive integers. The way to approach this successfully is what they call Vieta Jumping.
See what you can do.
On
The following Haskell script searches exhaustively for integer solutions in $[0,100]^4$:
xs = [0..100]
ys = [0..100]
zs = [0..100]
ws = [0..100]
p :: (Integer,Integer,Integer,Integer) -> Bool
p (x,y,z,w) = (x*y*z*w==504*(x^2+y^2+z^2+w^2))
sols :: [(Integer,Integer,Integer,Integer)]
sols = filter p [ (x,y,z,w) | x <- xs, y <- ys, z <- zs, w <- ws ]
Running the script, we obtain lots of integer solutions:
λ sols
[(0,0,0,0),(21,63,84,84),(21,72,75,75),(21,75,72,75),(21,75,75,72),(21,84,63,84),(21,84,84,63),(24,63,66,69),(24,63,69,66),(24,66,63,69),(24,66,69,63),(24,69,63,66),(24,69,66,63),(26,54,62,91),(26,54,91,62),(26,62,54,91),(26,62,91,54),(26,91,54,62),(26,91,62,54),(28,42,84,98),(28,42,98,84),(28,84,42,98),(28,84,98,42),(28,98,42,84),(28,98,84,42),(29,51,56,69),(29,51,69,56),(29,54,56,61),(29,54,61,56),(29,56,51,69),(29,56,54,61),(29,56,61,54),(29,56,69,51),(29,61,54,56),(29,61,56,54),(29,69,51,56),(29,69,56,51),(30,42,69,90),(30,42,90,69),(30,69,42,90),(30,69,90,42),(30,90,42,69),(30,90,69,42),(31,39,76,84),(31,39,84,76),(31,76,39,84),(31,76,84,39),(31,84,39,76),(31,84,76,39),(33,42,57,72),(33,42,72,57),(33,57,42,72),(33,57,72,42),(33,72,42,57),(33,72,57,42),(34,42,57,62),(34,42,62,57),(34,57,42,62),(34,57,62,42),(34,62,42,57),(34,62,57,42),(36,36,63,81),(36,36,81,63),(36,63,36,81),(36,63,81,36),(36,81,36,63),(36,81,63,36),(37,41,54,56),(37,41,56,54),(37,54,41,56),(37,54,56,41),(37,56,41,54),(37,56,54,41),(39,31,76,84),(39,31,84,76),(39,76,31,84),(39,76,84,31),(39,84,31,76),(39,84,76,31),(41,37,54,56),(41,37,56,54),(41,54,37,56),(41,54,56,37),(41,56,37,54),(41,56,54,37),(42,28,84,98),(42,28,98,84),(42,30,69,90),(42,30,90,69),(42,33,57,72),(42,33,72,57),(42,34,57,62),(42,34,62,57),(42,42,42,63),(42,42,42,84),(42,42,63,42),(42,42,84,42),(42,57,33,72),(42,57,34,62),(42,57,62,34),(42,57,72,33),(42,62,34,57),(42,62,57,34),(42,63,42,42),(42,69,30,90),(42,69,90,30),(42,72,33,57),(42,72,57,33),(42,84,28,98),(42,84,42,42),(42,84,98,28),(42,90,30,69),(42,90,69,30),(42,98,28,84),(42,98,84,28),(51,29,56,69),(51,29,69,56),(51,56,29,69),(51,56,69,29),(51,69,29,56),(51,69,56,29),(54,26,62,91),(54,26,91,62),(54,29,56,61),(54,29,61,56),(54,37,41,56),(54,37,56,41),(54,41,37,56),(54,41,56,37),(54,56,29,61),(54,56,37,41),(54,56,41,37),(54,56,61,29),(54,61,29,56),(54,61,56,29),(54,62,26,91),(54,62,91,26),(54,91,26,62),(54,91,62,26),(56,29,51,69),(56,29,54,61),(56,29,61,54),(56,29,69,51),(56,37,41,54),(56,37,54,41),(56,41,37,54),(56,41,54,37),(56,51,29,69),(56,51,69,29),(56,54,29,61),(56,54,37,41),(56,54,41,37),(56,54,61,29),(56,61,29,54),(56,61,54,29),(56,69,29,51),(56,69,51,29),(57,33,42,72),(57,33,72,42),(57,34,42,62),(57,34,62,42),(57,42,33,72),(57,42,34,62),(57,42,62,34),(57,42,72,33),(57,62,34,42),(57,62,42,34),(57,72,33,42),(57,72,42,33),(61,29,54,56),(61,29,56,54),(61,54,29,56),(61,54,56,29),(61,56,29,54),(61,56,54,29),(62,26,54,91),(62,26,91,54),(62,34,42,57),(62,34,57,42),(62,42,34,57),(62,42,57,34),(62,54,26,91),(62,54,91,26),(62,57,34,42),(62,57,42,34),(62,91,26,54),(62,91,54,26),(63,21,84,84),(63,24,66,69),(63,24,69,66),(63,36,36,81),(63,36,81,36),(63,42,42,42),(63,66,24,69),(63,66,69,24),(63,69,24,66),(63,69,66,24),(63,81,36,36),(63,84,21,84),(63,84,84,21),(66,24,63,69),(66,24,69,63),(66,63,24,69),(66,63,69,24),(66,69,24,63),(66,69,63,24),(69,24,63,66),(69,24,66,63),(69,29,51,56),(69,29,56,51),(69,30,42,90),(69,30,90,42),(69,42,30,90),(69,42,90,30),(69,51,29,56),(69,51,56,29),(69,56,29,51),(69,56,51,29),(69,63,24,66),(69,63,66,24),(69,66,24,63),(69,66,63,24),(69,90,30,42),(69,90,42,30),(72,21,75,75),(72,33,42,57),(72,33,57,42),(72,42,33,57),(72,42,57,33),(72,57,33,42),(72,57,42,33),(72,75,21,75),(72,75,75,21),(75,21,72,75),(75,21,75,72),(75,72,21,75),(75,72,75,21),(75,75,21,72),(75,75,72,21),(76,31,39,84),(76,31,84,39),(76,39,31,84),(76,39,84,31),(76,84,31,39),(76,84,39,31),(81,36,36,63),(81,36,63,36),(81,63,36,36),(84,21,63,84),(84,21,84,63),(84,28,42,98),(84,28,98,42),(84,31,39,76),(84,31,76,39),(84,39,31,76),(84,39,76,31),(84,42,28,98),(84,42,42,42),(84,42,98,28),(84,63,21,84),(84,63,84,21),(84,76,31,39),(84,76,39,31),(84,84,21,63),(84,84,63,21),(84,98,28,42),(84,98,42,28),(90,30,42,69),(90,30,69,42),(90,42,30,69),(90,42,69,30),(90,69,30,42),(90,69,42,30),(91,26,54,62),(91,26,62,54),(91,54,26,62),(91,54,62,26),(91,62,26,54),(91,62,54,26),(98,28,42,84),(98,28,84,42),(98,42,28,84),(98,42,84,28),(98,84,28,42),(98,84,42,28)]
One positive integer solution with distinct components is $(24,63,66,69)$.
On
The solution with the smallest maximum is $56;$
56 54 41 37
61 56 54 29
62 57 42 34
63 42 42 42
69 56 51 29
69 66 63 24
72 57 42 33
75 75 72 21
81 63 36 36
84 42 42 42
84 76 39 31
84 84 63 21
90 69 42 30
91 62 54 26
98 84 42 28
Next, if we fix two of the numbers, call them $W = 56, Z = 54,$ we get an indefinite binary quadratic form in $x,y$ that has infinitely many solutions. It works best when $WZ$ is a multiple of $504,$ because then we get a Vieta Jumping situation: $$ 56 \cdot 54 = 3024 = 504 \cdot 6. $$ From $wzxy = 504(w^2 + z^2 + x^2 + y^2)$ we have $$ 6 \cdot 504 \cdot xy = 504 (56^2 + 54^2 + x^2 + y^2), $$ $$ 6xy = 56^2 + 54^2 + x^2 + y^2, $$ $$ x^2 - 6 xy + y^2 = -6052. $$ We see that solutions require $x,y>0,$ and we already have $41, 37.$ It is a exercise in Vieta Jumping to display all the solutions for $ x^2 - 6 xy + y^2 = -6052. $ They come in two doubly infinite Vieta orbits, $$ (6113,1049); \; (1049, 181); \; (181,37); \; (37,41); \; (41,209); \; (209,1213), $$ $$ (3781,649); \; (649, 113); \; (113,29); \; (29,61); \; (61,337); \; (337,1961), $$
I told the machine to sort by $x+y$
x+y
78 x: 37 y: 41
78 x: 41 y: 37
90 x: 29 y: 61
90 x: 61 y: 29
142 x: 113 y: 29
142 x: 29 y: 113
218 x: 181 y: 37
218 x: 37 y: 181
250 x: 209 y: 41
250 x: 41 y: 209
398 x: 337 y: 61
398 x: 61 y: 337
762 x: 113 y: 649
762 x: 649 y: 113
1230 x: 1049 y: 181
1230 x: 181 y: 1049
1422 x: 1213 y: 209
1422 x: 209 y: 1213
2298 x: 1961 y: 337
2298 x: 337 y: 1961
4430 x: 3781 y: 649
4430 x: 649 y: 3781
7162 x: 1049 y: 6113
7162 x: 6113 y: 1049
8282 x: 1213 y: 7069
8282 x: 7069 y: 1213
13390 x: 11429 y: 1961
13390 x: 1961 y: 11429
25818 x: 22037 y: 3781
25818 x: 3781 y: 22037
41742 x: 35629 y: 6113
41742 x: 6113 y: 35629
48270 x: 41201 y: 7069
48270 x: 7069 y: 41201
78042 x: 11429 y: 66613
78042 x: 66613 y: 11429
150478 x: 128441 y: 22037
150478 x: 22037 y: 128441
243290 x: 207661 y: 35629
243290 x: 35629 y: 207661
281338 x: 240137 y: 41201
281338 x: 41201 y: 240137
454862 x: 388249 y: 66613
454862 x: 66613 y: 388249
x + y
What happens when $WZ$ is not a multiple of $504?$ It is similar, but not directly what you know as Vieta Jumping. It is still, however, a collection of orbits under the automorphism group of a quadratic form.
Fix two of the numbers, call them $W = 56, Z = 69,$ we get an indefinite binary quadratic form in $x,y$ that has infinitely many solutions. $$ 56 \cdot 69 = 3864 = 168 \cdot 23. $$ Here $$ 168 = \gcd(504, WZ), $$ From $wzxy = 504(w^2 + z^2 + x^2 + y^2)$ we have $$ 23 \cdot 168 \cdot xy = 3 \cdot 168 (56^2 + 69^2 + x^2 + y^2), $$ $$ 23xy = 3 \left( 56^2 + 69^2 + x^2 + y^2 \right), $$ $$ 3x^2 - 23 xy + 3y^2 = -23691. $$ We see that solutions require $x,y>0,$ and we already have $51, 29.$
In the output below, the recipe that takes $(x,y)$ to "forward" is $$ (x,y) \mapsto ( 12544 x -1665 y, 1665 x -221 y) $$
3 x^2 - 23 x y + 3 y^2 = -23691
x 29 y 51 forward 278861 37014 backward 78506 591459
x 51 y 29 forward 591459 78506 backward 37014 278861
x 246 y 37 forward 3024219 401413 backward 7239 54538
x 37 y 246 forward 54538 7239 backward 401413 3024219
x 362 y 51 forward 4456013 591459 backward 4913 37014
x 51 y 362 forward 37014 4913 backward 591459 4456013
x 1849 y 246 forward 22784266 3024219 backward 961 7239
x 246 y 1849 forward 7239 961 backward 3024219 22784266
x 7239 y 961 forward 89205951 11840554 backward 246 1849
x 961 y 7239 forward 1849 246 backward 11840554 89205951
x 37014 y 4913 forward 456123471 60542537 backward 51 362
x 4913 y 37014 forward 362 51 backward 60542537 456123471
x 54538 y 7239 forward 672071737 89205951 backward 37 246
x 7239 y 54538 forward 246 37 backward 89205951 672071737
x 278861 y 37014 forward 3436404074 456123471 backward 29 51
x 37014 y 278861 forward 51 29 backward 456123471 3436404074
x 591459 y 78506 forward 7288549206 967429409 backward 51 29
x 78506 y 591459 forward 29 51 backward 967429409 7288549206
x 3024219 y 401413 forward 37267450491 4946612362 backward 246 37
x 401413 y 3024219 forward 37 246 backward 4946612362 37267450491
x 4456013 y 591459 forward 54911447837 7288549206 backward 362 51
x 591459 y 4456013 forward 51 362 backward 7288549206 54911447837
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Wow, amazing amount of work. I went through and collected the solutions above, and more, into orbits. There are eight $SL_2 \mathbb Z$ orbits. Since there is an obvious involution, switching $x$ and $y,$ and that linear transformation has determinant $-1,$ we could also say that there are four $GL_2 \mathbb Z$ orbits. I did check the Hurwitz type inequalities, this is very similar to what Vieta Jumping gives you. For this problem, $3 x^2 - 23 xy + 3 y^2 = - 23691,$ for any $SL_2 \mathbb Z$ orbit of solutions with both $x,y$ positive, there is a solution with $\color{red}{x + y < 4144.39}$ in that orbit.
12323^2 - 493 555^2 = 4
3 x^2 + -23 x y 3 y^2 = -23691
$$ (967429409 , 7288549206); \; \; ( 78506 591459); \; \; (29, 51); \; \; (278861, 37014); \; \; (3436404074, 456123471); $$
$$ ( 456123471, 3436404074 ); \; \; ( 37014, 278861); \; \; ( 51, 29); \; \; (591459, 78506); \; \; ( 7288549206, 967429409); $$
$$ $$
$$ ( 89205951, 672071737 ); \; \; ( 7239, 54538 ); \; \; (246, 37); \; \; (3024219, 401413); \; \; ( 37267450491, 4946612362 ); $$
$$ ( 4946612362, 37267450491 ); \; \; (401413, 3024219); \; \; (37, 246); \; \; (54538, 7239); \; \; (672071737, 89205951 ); $$
$$ $$
$$ ( 746065678538, 5620809496119 ); \; \; ( 60542537, 456123471 ); \; \; ( 4913, 37014 ); \; \; ( 362, 51); \; \; (4456013, 591459); \; \; ( 54911447837, 7288549206 ); $$
$$ ( 7288549206, 54911447837 ); \; \; ( 591459, 4456013 ); \; \; (51, 362); \; \; (37014, 4913); \; \; ( 456123471, 60542537 ); \; \; ( 5620809496119, 746065678538 ); $$
$$ $$
$$ ( 145911145981, 1099284926934 ); \; \; ( 11840554, 89205951 ); \; \; ( 961, 7239 ); \; \; (1849, 246); \; \; (22784266, 3024219); \; \; ( 280770508069, 37267450491 ); $$
$$ ( 37267450491, 280770508069 ); \; \; ( 3024219, 22784266 ); \; \; ( 246, 1849); \; \; (7239, 961); \; \; ( 89205951, 11840554 ); \; \; ( 1099284926934, 145911145981 ); $$
$$ $$
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Using a Pell-like equation, there are in fact infinitely many positive integer solutions to,
$$xyzw=504(x^2+y^2+z^2+w^2)\tag1$$
$$\big(x,y,z,w\big)=\big(84,\;84,\;21q,\;21(4p+7q)\big)$$
where, $$p^2-3q^2=-2\tag2$$
An initial point is $(p,q) = (-1,1)$ yielding the OP's known $\big(x,y,z,w\big)=\big(84,\;84,\;21,\;63\big)$. As $(2)$ has infinitely many positive integer solutions, then so does $(1)$.
$$\big(x,y,z,w\big)=\big(36,\;36,\;9(4p+5q),\;9(8p+11q)\big)$$
where, $$p^2-2q^2=14\tag3$$
An initial point is $(p,q) = (4,-1)$ yielding $\big(x,y,z,w\big)=\big(36,\;36,\;99,\;189\big)$. And infinitely more.
(Added a day later.)
$$\big(x,y,z,w\big)=\big(12c,\;12c,\;c(ap+7bq-ac^2q),\;c(bp-7aq+bc^2q)\big)$$
with the Pell-like,
$$\beta\, p^2-\beta(c^4-49)\,q^2=7\times 288=\color{blue}{2016}$$
where $\beta=-7 a^2 - 7 b^2 + 2 a b c^2$ for arbitrary $a,b,c$ and we recover the $2016$ asked by the OP. The first two families were just special cases.