Can you calculate Pi using a Solver (on the prime)?

[Albert Chan]

This shows (on XCas) that FNROOT can not be nested

XCas>FNF(v):={local FVAR; purge(FVAR); print(v); FNROOT(Gamma(FVAR)=v,FVAR=0..1)[0]}

XCas>m := makelist(FNF, 1.75, 1.8, 0.01)
v:1.75
v:1.76
v:1.77
v:1.78
v:1.79
v:1.8
→ [0.506546500849,0.503607372216,0.500706197505,0.497842178366,0.495014540318,0.492​222531811]

XCas>map(m, Gamma)                          // confirm FNF = Gamma^-1, at least for domain [1.75,1.8]
→ [1.75,1.76,1.77,1.78,1.79,1.8]

XCas>FNROOT(FNF(sqrt(x))=0.5, x=3)   // unable to get value of sqrt(x) into FNF
v:sqrt(x)
"Error: Invalid dimension"