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})`

Thanks in advance.