Proof that 10 is Solitary
Dan Heflin
What is a Solitary Number?
• A solitary number is a number that has no
friendly pair. That is, a solitary number is in a
singleton group, with only itself.
What is a Friendly Pair?
• A friendly pair is a pair of numbers that has
the same ratio of the sum of divisors to the
number itself.
• So, for example, 6 and 28 are friendly.
– 6 has divisors 1,2,3,6; 1+2+3+6 = 12/6 = 2
– 28 has divisors 1,2,4,7,14,28; 1+2+4+7+14+28 =
56/28 = 2
Friendly Numbers
• 6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84,
96, 102, 108, 114, 120, 132, 135, 138, 140,
150, 168, 174, 186, 200, 204, 210, 222, 224,
228, 234, 240, 246, 252, 258, 264, 270, 273,
276, 280, 282, 294, 300, 308, 312, 318, 330,
348, 354, 360, 364, 366, 372
• http://joelbradbury.net/notes/friendly_numb
ers
Solitary Numbers
• These numbers include all prime numbers, all
powers of prime numbers, and all other numbers
where the ratio of the sum of the divisors to the
number itself matches no other ratio.
• 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196,
208, 232, 244, 261, 272, 292, 296, 297, 304, 320,
352, and 369 are all solitary numbers.
• http://mathworld.wolfram.com/SolitaryNumber.
html
Is 10 a Solitary Number?
• Well, Mathematicians all around the world
have wondered if 10 was a solitary number, or
if it has a friend out there somewhere.
• It is also the smallest number where people
are unsure of whether or not it has a friend.
• 10 must either be a solitary number, or its
friend has a very large index.
Our Proposition
• The question at hand has been pondered for
many years throughout Mathematical history.
• The problem is, 10 is not a power of a prime,
nor is it prime itself.
• This means that in order to prove that it is
indeed solitary, it must be disproved that it is
friendly, or we must find a way to prove a
number as being solitary.
However
• No Mathematician has been able to prove or
disprove this theory.
• In order to do this, Mathematicians would
have to discover a way to find solitary
numbers, or discover a way to find friendly
numbers.
• But, no one has been able to do this.
The End
• As a class we could sit down for a long time
and try to find 10 a friend, but I think we
would all agree not to do that!