Find the maximum integer such that this algebraic constraint is satisfied in Mathematica

Question

For example, I have an expression as a (arbitrary) function of integer n

f[n_]:=10^n*(n^2+4*n)

I want to find the maximum integer n such that f[n]<=m for another number m.

I could formulate this as an integer programming/optimization problem. But that complicates things. I could also just try starting from 1 and continue to test whether the constraint is violated. Is there any more efficient or elegant way of doing this? Please note also the constraint may allow an Infinity value of n and I ideally want to detect this situation.

If x is real, can one assume f[x] is a continuous function? – Arnoud Buzing Nov 20 '11 at 02:40 — Arnoud Buzing, Nov 20 '11 at 02:40
yes, you can assume f[x] is continuous. – Qiang Li Nov 20 '11 at 17:00 — Qiang Li, Nov 20 '11 at 17:00

score 5 · Answer 1 · answered Nov 19 '11 at 06:27

Depends. If you can settle for a heuristic result using numeric methods, that makes the assumption an integer max is the floor of a real max, then can do as below.

f[n_] := 10^n*(n^2 + 4*n)

In[32]:= Floor[First[NMaximize[{n, f[n] <= 10^8}, n]]]
Out[32]= 6

Daniel Lichtblau

Find the maximum integer such that this algebraic constraint is satisfied in Mathematica

1 Answers1