# Reduce does not get a result

I try to solve the following inequation.

Reduce[
(k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n)) < 1/n &&
x > 0 && y > 0 && n ∈ Integers &&k ∈ Integers,
{x, y, k, n}]


Unfortunately,Reduce does not get a result. Any suggestion on how to solve it?

• Although the documentation does not explicitly mention it, Reduce[] is very lame when dealing with exponentials. For example, it can't do Reduce[2 == 2^n + 2^m, {n, m}, Integers] which should be easy. Try FindInstance[] instead which is more capable. – Somos Mar 30 at 18:28
• As I read this, you are trying to find a solution for 4 variables from one equation. Is this what you want? Do you really want an inequality for x? – mikado Mar 30 at 18:28
• Something can be done by specification of $n$ and $k$, e.g. Reduce[(n*((k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n))) < 1) /. {n -> 1, k -> 2} && x > 0 && y > 0, {x, y}, Reals] – user64494 Mar 30 at 18:31
• If you are looking for an upper bound, please place that important informaion in the body of your question instead of a comment. – Somos Mar 30 at 22:27
• By the way, the inequality is false in general. Try n=k=x=y=2. – Somos Mar 30 at 22:38

Although the answer provided by bill-s provides ten instances violating the original inequality,

FindInstance[(k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n)) < 1/n
&& x > 0 && y > 0 && n \[Element] Integers && k \[Element] Integers, {x, y, k, n}, 10]


gives ten instances satisfying the original inequality. An example is

(* {x -> 5683/5, y -> 2331/10, k -> 10084, n -> 1132} *)
((k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n)) < 1/n) /. %
(* True *)


It is, therefore, natural to ask what portion of random sets of {x, y, k, n} satisfy the inequality. The expression,

Count[ParallelTable[{x = RandomReal[10^5], y = RandomReal[10^5],
k = RandomInteger[{-10^5, 10^5}], n = RandomInteger[{-10^5, 10^5}],
tst = (k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n)),
tst < 1/n}, 100000], False, Infinity]


suggests that about 70% do. Replacing the right side of the inequality to 2/n increases the percentage to about 82%, but further increasing the right side moderately yields no improvement. An example of a random set for which the inequality is far from satisfied is

(* {99896.2, 4488.5, -40330, -43962, 7.317356679375576*10^202473, False} *)


Such instances arise when k and n both are large negative integers, with k or n even, and also occasionally when k is positive and n is negative and even. Further, the left side of inequality can be arbitrarily large amount in such instances.

To assure that the inequality always is satisfied, require that both k > 1 and n > 0, and increase the right side of the inequality to 2/n. (A tighter bound, say 1.6/n, also works.)

Taking a hint from Somos, you can see that your inequality is false in general. Use FindInstance with the opposite sign on the inequality:

FindInstance[(k^-n (k (-k + k^n) y + (-1 + k) x (-1 + k + y)))/((-1 + k) (x + y*n)) > 1/n
&& x > 0 && y > 0 && n \[Element] Integers && k \[Element] Integers, {x, y, k, n}, 10]


This returns 10 cases where the original inequality, with less then 1/n, is false.