Am I missing any parameter's to get a solution

Questions
,
1

I am trying to solve an equation that I know has solutions, but what I am trying isn’t working as expected, am I missing something? Here is the equation

Reduce[{160755614715009600 + 152 y^2 + 2630 z^2 == 
   m (2567290440 + 152 y + 2630 z), y > 0, z > 0, m > 0}, {y, z, 
  m}, Integers]

Which gives me

(y | z | m) \[Element] Integers && y >= 1 && z >= 1 && 
 m == (80377807357504800 + 76 y^2 + 1315 z^2)/(
  1283645220 + 76 y + 1315 z)

Yet if I give the value for “m”, I get a solution thus

Reduce[{160755614715009600 + 152 y^2 + 2630 z^2 == 
   33395648 (2567290440 + 152 y + 2630 z), y > 0, z > 0}, {y, 
  z}, Integers]

Resulting in

(y == 37928418 && z == 1194380) || (y == 37928418 && 
   z == 32201268) || (y == 46962630 && 
   z == 2087228) || (y == 46962630 && z == 31308420)

Or am I expecting too much?

2

Hi Paul,

Interesting that FindInstance gives this solution

eq = 160755614715009600 + 152 y^2 + 2630 z^2 - m (2567290440 + 152 y + 2630 z) == 0
FindInstance[{eq, y > 0, z > 0, m > 0}, {m, y, z}, Integers]
(* {{m -> 62616840, y -> 62616840, z -> 62616840}} *)

eq /. {m -> 62616840, y -> 62616840, z -> 62616840}
(* True *)
3

It’s interesting that FindInstance only finds 1 solution, when there are many solutions, these are just the first few

62616840 , {{y->62616840,z->62616840}}
    62616826 , {{y->131915080,z->10050766},{y->131915080,z->52566060}}
    62616812 , {{y->92362065,z->2647718},{y->92362065,z->59969094}}
    62616758 , {{y->122326338,z->7684682},{y->122326338,z->54932076}}
    62616688 , {{y->25532718,z->63478756},{y->27384238,z->63494884},{y->35232450,z->63494884},{y->37083970,z->63478756},{y->96758670,z->3214116},{y->96758670,z->59402572}}

Maybe its because there’s too many solutions.