Find all valid triplets in this equation

I'm trying to find all positive integer triplets a,b,c that satisfy this equation. Note r is a constant. And 1 does exist: 23,13,121. Why is the result not found? Is there something wrong with the way I expressed the first expression?

For those with older versions of Mathematica, please replace "PositiveIntegers" with "Integers".

eqn1 = FullSimplify[{2*r^98 + 14*r^96 + 11*r^94 - r^50 + a*r^46 + b*r^44 + c*r^40 == r^100, r = Sqrt[(Sqrt[53]/2) + 1.5] }]

Out: {2.25039*10^16 a + 4.37815*10^15 b + 1.65712*10^14 c == 0, 2.26717}

Table[FindInstance[eqn1, {a, b, c}, PositiveIntegers, 15]]

Out: FindInstance[{2.25039*10^16 a + 4.37815*10^15 b + 1.65712*10^14 c == 0, 2.26717}, {a, b, c}, PositiveIntegers, 15]

Also: can it be solved outright?

Solve[{2*r^98 + 14*r^96 + 11*r^94 - r^50 + a*r^46 + b*r^44 + c*r^40 == r^100, r = Sqrt[(Sqrt[53]/2) + 1.5]}, {a, b, c}]

eqn1 = FullSimplify[
(2*r^98 + 14*r^96 + 11*r^94 - r^50 + a*r^46 + b*r^44 + c*r^40 ==
r^100) /. r -> Sqrt[(Sqrt[53]/2) + 3/2]]


FindInstance[eqn1, {a, b, c}, Integers, 15] // Column


But, with PositiveIntegers as the domain, FindInstance can find only one solution:

FindInstance[eqn1, {a, b, c}, PositiveIntegers, 15]

{{a -> 23, b -> 13, c -> 121}}


Using Solve:

Solve[eqn1, {a, b, c}, Integers]


Solve[eqn1, {a, b, c}, PositiveIntegers]

{{a -> 23, b -> 13, c -> 121}}

• Thanks @kglr Is this the only part you changed? "/. r ->" Did the original method had a syntax or wrong way to represent? Also: Why isn't the 23,13,121 showing up in the Integers version? – Steve237 Feb 22 at 15:50
• I get your first set, but my FindInstance[eqn1, {a, b, c}, PositiveIntegers, 15] returns nothing. neither does the Solve. pasted as-is. Solve says "Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result." ... – Steve237 Feb 22 at 15:54
• Hi @kglr. See my output! I am not getting even the 15 solutions you have above, they are different. [my output] (drive.google.com/file/d/1l1745GnGH90LtTqqtmjqsnLwwel3OxwW/view). Why is this? is a setting incorrect on my system? – Steve237 Feb 22 at 16:11
• @Steve237, I also changed 1.5 to 3/2. – kglr Feb 22 at 16:25
• Never mind, I closed down the tool and re-opened fresh and it works. Not sure if it was using references to the previous inputs. – Steve237 Feb 22 at 19:04