## 10.3.11

### Some Open Problems on Number Theory

 Solitary Number
A number which does not have any friends. Solitary numbers include all primes,prime powers, and numbers for which , where  is the greatest common divisor of  and  and  is the divisor function. The first few numbers satisfying  are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (Sloane'sA014567). Numbers such as 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 can also be easily proved to be solitary (Hickerson 2002).

Some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large. For example, 24 is friendly because  is a friendly pair. However, there exist numbers such as , 45, 48, and 52 which are solitary but for which . It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary, although a proof appears to be extremely difficult.

In 1996, Carl Pomerance told Dean Hickerson that he could prove that the solitary numbers have positive density, disproving a conjecture by Anderson and Hickerson (1977). However, this proof seems not to ever have been published (Hickerson 2002).

# Friendly number

In number theory, a friendly number is a natural number that shares a
certain characteristic called abundancy, the ratio between the sum of
divisors of the number and the number itself, with one or more other
numbers. Two numbers with the same abundancy form a friendly pair.

Being mutually friendly is an equivalence relation, and thus induces a
partition of the positive naturals into clubs (equivalence classes) of
mutually friendly numbers.

A number that is not part of any friendly pair is called solitary.

The abundancy of n is the rational number σ(n) / n, in which σ denotes
the sum of divisors function. A number n is a friendly number if there exists
m ≠ n such that σ(m) / m = σ(n) / n. Note that abundancy is not the same as
abundance which is defined as σ(n) − 2n.

Abundancy may also be expressed as $\sigma_{-\!1}(n)$ where σk denotes a divisor
function with σk(n) equal to the sum of the k-th powers of the divisors of n.

The numbers 1 through 5 are all solitary. The smallest friendly number is 6,
forming for example the friendly pair (6, 28) with abundancy
σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as
σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in
this case but not in many other cases. There are several unsolved problems
related to the friendly numbers.

In spite of the similarity in name, there is no specific relationship between the
friendly numbers and the amicable numbers or the sociable numbers
although the definitions of the latter two also involve the divisor function.

## Solitary numbers

A number that belongs to a singleton club, because no other number is friendly with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so
that σ(n)/n is an irreducible fraction – then the number n is solitary. For a prime number p we have σ(p) = p+ 1, which is coprime with p.

No general method is known for determining whether a number is friendly or solitary. The smallest number whose classification is unknown (as of 2009) is 10;  it is conjectured to be solitary; if not, its smallest friend is a fairly large number.  [Refer to this arXiv article for more information.]

Open Problem:  Are all (odd) squares solitary?

See initial investigations here:

Related post over at MathOverflow here:

#### 1 comment:

Arnie Dris said...

>During the course of my research, I came
>across the concept of solitary numbers, and >this link on OEIS (http://oeis.org/A014567).

>That last link lists the following as >solitary numbers:
>1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, >21, 23, 25, 27, 29, 31, 32, 35, 36, 37, 39, >41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, >64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, >89, 93, 97, 98, 100, 101, 103, 107, 109, 111, >113, 115, 119, 121, 125, 127, 128, 129, 131, >133

>Notice that all of the squares from 1 to 121 >are solitary (since they satisfy
>gcd(n, sigma(n)) = 1).

>Does my observation hold true in general? In >particular, is there a natural number m such >that m = k^2 and gcd(m, sigma(m)) is not >equal to 1?

Reply from personal communication with Dean Hickerson:

No. The smallest square for which gcd(n, sigma(n)) is not equal to 1 is 196: gcd(196, sigma(196)) = 7. (But it's easy to show that 196 is solitary.)

In 1995 I found a square that isn't solitary; I don't know if there are any smaller ones: 26334^2 = 693479556 = 2^2 3^4 7^2 11^2 19^2. There are at least 5 other numbers with the same abundancy index:

8640 = 2^6 3^3 5
52416 = 2^6 3^2 7 13
71814642425856 = 2^13 3^4 11^3 31 43 61
2168446760665473024 = 2^13 3^10 11 23 43 107 3851
5321505362711814144 = 2^13 3^6 11 23 43 137 547 1093

> I had expected my conjecture to fail for
> even squares. Notwithstanding, have you
> also found any odd squares which are not
> solitary?

No, I haven't. It's easy to find odd squares for which gcd(n, sigma(n)) is not equal to 1; e.g. if n = 21^2 = 441 then gcd(n, sigma(n)) = 3. But 441 is solitary.