Stealthy Numbers

A positive integer $N$ is stealthy, if there exist positive integers $a$, $b$, $c$, $d$ such that $ab = cd = N$ and $a+b = c+d+1$.
For example, $36 = 4\times 9 = 6\times 6$ is stealthy.

You are also given that there are 2851 stealthy numbers not exceeding $10^6$.

How many stealthy numbers are there that don’t exceed $10^{14}$?

The definition of stealthy numbers involves two equations with four variables $a$, $b$, $c$, $d$. To solve this, we can do:

We know that $ab=cd=N$, and that $a+b=c+d+1$.

Starting from this, we can write $a=N/b$, $c=(b-1)/2$, and $d=(b+1)/2$.

We replace the terms in $ab=cd=N$ and get:

$(N/b)*b=(b-1)/2 * (b+1)/2 = N$

After simplifying, this gives us:

$b^4-1=4N$, so $b^4=4N+1$.

Starting from this, we have to find the numbers $b$ that make $N=(b^4-1)/4$ a positive integer, and that $b$, $c$, $d$ are also integers.

We need to look for b values for which $(b-1)/2$ and $(b+1)/2$ are both integers, so basically b should be an odd number.

Also, for the stealthy number $N$ to not exceed $10^{14}$, b should not exceed the fourth root of $(10^{14}+1)$, which is approximately $3162.2776$ considering only integer values.

So we have to look for positive odd integer numbers starting from $b=1$ up to $b=3163$, and calculate $N=(b^4-1)/4$. If $N$ is an integer, we have a stealthy number.

Doing this, we obtain 7549747 stealthy numbers not exceeding $10^{14}$.

More Answers: