Only Number which is Sum of 3 Factors is 6
Theorem
The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors is $6$.
Corollary
$1$ can be expressed uniquely as the sum of $3$ distinct unit fractions:
- $1 = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 6$
Proof
Let $n$ be such a positive integer with corresponding divisors $a, b, c$ such that:
- $a + b + c = n$
We note that the set $\set {k a, k b, k c}$ satisfy the same properties trivially as divisors of $k n$.
Hence the specification that $\set {a, b, c}$ is a coprime set.
Without loss of generality, suppose $a < b < c$.
Since $a, b, c$ are strictly positive, $n \ne c$.
Suppose $\dfrac n c \ge 3$.
Then:
\(\ds n\) | \(=\) | \(\ds a + b + c\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds c + c + c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n\) |
which is a contradiction.
Hence:
\(\ds \frac n c\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + b + c\) | \(=\) | \(\ds 2 c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + b\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n\) | \(=\) | \(\ds 2 a + 2 b\) |
Since $a, b$ are divisors of $n$:
- $a \divides \paren {2 a + 2 b}$
- $b \divides \paren {2 a + 2 b}$
which reduces to:
- $a \divides 2 b$
- $b \divides 2 a$
Suppose $b$ is odd.
Then by Euclid's Lemma, we would have $b \divides a$.
By Absolute Value of Integer is not less than Divisors, this gives $b \le a$, which is a contradiction.
Thus $b$ is even.
Suppose $a$ is even.
Then $a, b, c$ are all even.
So $\gcd \set {a, b, c} \ne 1$, which is a contradiction.
Therefore it must be the case that $a$ is odd.
Then by Euclid's Lemma, we have:
- $a \divides \dfrac b 2$
and:
- $\dfrac b 2 \divides a$
By Absolute Value of Integer is not less than Divisors, this gives:
- $\dfrac b 2 = a$
Because $\gcd \set {a, b, c} = 1$, we must have $a = 1$.
Hence the set $\set {1, 2, 3}$ is obtained.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$