Minimizing simple function of three variables fails

Question

I need to minimize simple function with all variables are positive integers, but the out is the same as the input. No solution
    Clear[α, k, β, g];
g = (1 + Sqrt[5])/2;
f[α_, k_, β_, g_] := Abs[π - (α/β)*g^k;]
Assuming[α > 0 && k > 0 && β > 0 , 
 Minimize[f[α, k, β, g], {α, k, β}, 
  Integers]]

My version is 12.1
Any help
Thanks

Ulrich Neumann · Answer

First correct the definition of f (semicolon inside argument brackets) g = (1 + Sqrt[5])/2; f[[Alpha]_, k_, [Beta]_, g_] := Abs[[Pi] - ([Alpha]/[Beta])*g^k] Use NMinimize with constraints instead of assumptions: NMinimize[{f[[Alpha], k, [Beta], g],Element[{[Alpha], k, [Beta]}, PositiveIntegers]}, {[Alpha],k, [Beta]} , MaxIterations -> 500] (*{ 0.0354583, {[Alpha] -> 3, k -> 3, [Beta] -> 4}}*) addendum Restricting the parameterrange {[Alpha] -<10, k<10, [Beta] <10} helps NMinimize finding the "right" minimum NMinimize[{f[[Alpha], k, [Beta], g], [Alpha] < 10, k < 10, [Beta] < 10,Element[{[Alpha], k, [Beta]},PositiveIntegers]}, {[Alpha],k, [Beta]} , MaxIterations -> 500] (*{0.0000481329, {[Alpha] -> 6, k -> 2, [Beta] -> 5}}*)

user64494 · Answer

The infimum under consideration equals zero. Put k=1, then you approximate a number Pi/g==2*Pi/(1+Sqrt[5]) by a rational number [Alpha]/[Beta]. This can be done with an arbitrary accuracy, e.g.
RealAbs[FromContinuedFraction[ContinuedFraction[(2 [Pi])/(1 + Sqrt[5]), 
 20]] - (2 [Pi])/(1 + Sqrt[5])] // N
(*2.22045*10^-16*)
FromContinuedFraction[ContinuedFraction[(2 [Pi])/(1 + Sqrt[5]),20]]
(*107070177060/55145018711*)

Minimizing simple function of three variables fails

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