Lonely number

Under a solitary number ( English solitary number ) is understood in the mathematical sub-area of number theory is a natural number which no other natural number as known has. Two natural numbers are considered to be known or known to each other if the quotients formed from the divisional sum of the number and the number itself are identical for both . Lone numbers include all prime numbers .

definition

A natural number is called a lonely number (or in short: lonely ) if and only if the following applies: ${\displaystyle n_{0}}$

${\displaystyle \forall n\in {\mathbb {N} \setminus \{n_{0}\}}\;\colon \;{\frac {\sigma (n_{0})}{n_{0}}}\neq {\frac {\sigma (n)}{n}}}$

${\displaystyle \sigma (n)}$is the partial sum of , i.e. the sum of all factors of . ${\displaystyle n}$ ${\displaystyle n}$

Examples and Notes

• Every natural number , which has no divisor in common with its divisor sum except for, for which the divisor sum and the number itself are prime , is a lone number. Therefore all prime numbers and even in general all prime number powers belong to the lonely numbers . ${\displaystyle n_{0}}$ ${\displaystyle \sigma (n_{0})}$ ${\displaystyle 1}$ ${\displaystyle \sigma (n_{0})}$ ${\displaystyle n_{0}}$
• No perfect number is lonely, because it always applies why all perfect numbers are known to one another. ${\displaystyle N}$ ${\displaystyle {\frac {\sigma (N)}{N}}=2}$
• Numbers belong to the natural numbers, which are proven to be lonely without them and their partial sums being prime . ${\displaystyle 18,45,48,52}$
• There are at least solitary numbers below . ${\displaystyle 100}$ ${\displaystyle 53}$
• The proof that a natural number has a friend and therefore cannot be a lonely number is often extremely time-consuming, even for small natural numbers. For example, the number as the smallest acquaintance has the number . ${\displaystyle 24}$ ${\displaystyle 91963648}$

assumptions

There is a previously unproven guess that the following numbers are lonely:

• ${\displaystyle 10,14,15}$
• ${\displaystyle 20,22,26}$
• ${\displaystyle 33,34,38}$
• ${\displaystyle 44,46}$
• ${\displaystyle 51,54,58}$
• ${\displaystyle 62,68,69}$
• ${\displaystyle 70,72,74,76}$
• ${\displaystyle 82,86,87,88}$
• ${\displaystyle 90,91,92,94,95,99}$
• ${\displaystyle 104,105,106}$

Another open problem is whether there are infinite sets of mutually known numbers. One possible candidate is the set of perfect numbers.

literature

• CW Anderson, Dean Hickerson, MG Greening: Problems and Solutions: Solutions of Advanced Problems: 6020 . In: Amer. Math. Monthly . tape 84 , 1977, pp. 65-66 , doi : 10.2307 / 2318325 , JSTOR : 2318325 ( MR1538261 ).
• Jörg Neunhäuserer : Nice sentences in mathematics. An overview with brief evidence . Springer Spectrum, Berlin / Heidelberg 2015, ISBN 978-3-642-54689-1 .
• József Sándor , Borislav Crstici : Handbook of Number Theory. II . Kluwer Academic Publishers, Dordrecht / Boston / London 2004, ISBN 1-4020-2546-7 ( MR2119686 ).

Web link

Wolfram MathWorld Solitary Number

Individual evidence

1. ↑ ^{a b c d} Neunhäuserer: pp. 186–187.
2. ↑ ^{a b} Sándor-Crstici: pp. 70–71.
3. ↑ In the English-language specialist literature, two different numbers that are known to one another are referred to as friendly pairs .
4. ↑ Numbers that are known to one another are to be distinguished from friendly numbers .
5. ↑ The proof of this goes back to MG Greening. See Anderson-Hickerson-Greening: Amer.Math.Monthly . S. 65-66 .
6. ↑ ^{a b} Follow A095739 in OEIS