Untouchable number

In number theory is an untouchable number (from the English untouchable number ) is a positive integer that is not considered sum of all the proper divisors of any positive integer can be represented (including the untouchable number itself). These numbers do not appear in any content chains . They were first examined by ʿAbd al-Qāhir al-Baghdādī (around the year 1000), who noticed that the two numbers 2 and 5 are untouchable. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \sigma ^{*}(k)}$ ${\displaystyle k\in \mathbb {N} }$

Examples

• The number is an untouchable number. ${\displaystyle n=5}$

Proof:

One has to show that the sum of the real divisors of any number cannot be represented. ${\displaystyle n=5}$ ${\displaystyle k\in \mathbb {N} }$

No divisor occurs more than once in a real divisor sum . There are only two possibilities, the number 5 additive show with different numbers: . The second representation is not a partial sum because the number 1 is missing, which must always be included in every partial sum. The first representation is also out of the question as a divisor sum, because if a number has the divisors 1 and 4, it must also have the number 2 as a divisor and therefore its divisor sum must be at least . Thus there is no possibility left that there is such a thing . The number 5 is therefore an untouchable number. ${\displaystyle \sigma ^{*}(k)}$ ${\displaystyle 5=1+4=2+3}$ ${\displaystyle 5=2+3}$ ${\displaystyle 1+2+4=7\not =5}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \sigma ^{*}(k)=5}$ ${\displaystyle \Box }$

• The number is not an untouchable number. ${\displaystyle n=4}$

Proof:

One has to show that the sum of the proper factors is any number . ${\displaystyle n=4}$ ${\displaystyle k\in \mathbb {N} }$

It is . There is a number that only has the numbers 1 and 3 as real divisors, namely the number . So its partial sum is , with which the number 4 is not an untouchable number. ${\displaystyle 4=1+3}$ ${\displaystyle k=9}$ ${\displaystyle \sigma ^{*}(9)=1+3=4}$ ${\displaystyle \Box }$

• The following numbers are the smallest untouchable numbers:

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658, ... (sequence A005114 in OEIS )

properties

• Perfect numbers are never untouchable numbers.
  Proof:
  • Be a perfect number. The property of perfect numbers is that they are equal to their real partial sum. So it applies . So there is a number whose real partial sum is equal . Thus is not an untouchable number. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \sigma ^{*}(n)=n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \Box }$
• Friends numbers are never untouchable numbers.
  Proof:
  • Be two numbers friend. Friendly numbers have the property that one number is equal to the real divisional sum of the other number and vice versa. So it is and . Thus for each of the two numbers and there is a real partial sum that is equal or is. Thus and are not intangible numbers. ${\displaystyle n_{1},n_{2}\in \mathbb {N} }$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle \sigma ^{*}(n_{1})=n_{2}}$ ${\displaystyle \sigma ^{*}(n_{2})=n_{1}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle \Box }$
• Social numbers are never untouchable numbers.
  Proof:
  • Be with sociable numbers. Sociable numbers have the property that the real divisional sum of the -th number is equal to (with ). So it is and . Thus for each of the numbers there is a true divisional sum that is the same . Thus there are no untouchable numbers. ${\displaystyle n_{1},n_{2},\ldots n_{k}\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} ,k\geq 3}$ ${\displaystyle i}$ ${\displaystyle n_{i}}$ ${\displaystyle n_{i+1}}$ ${\displaystyle i=1,2,\ldots k-1}$ ${\displaystyle \sigma ^{*}(n_{i})=n_{i+1}}$ ${\displaystyle \sigma ^{*}(n_{k})=n_{1}}$ ${\displaystyle k}$ ${\displaystyle n_{1},n_{2},\ldots n_{k}}$ ${\displaystyle n_{1},n_{2},\ldots n_{k}}$ ${\displaystyle n_{1},n_{2},\ldots n_{k}}$ ${\displaystyle \Box }$
• Let be a prime number and a number that is 1 greater than a prime number. Then the following applies: is not an untouchable number. Proof: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle n=p+1}$
  ${\displaystyle n}$
  
  • Because is a prime number, only has the real factors and . Thus applies to the real partial sum . So there can never be an untouchable number because there is a number whose real prime is the same . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle p^{2}}$ ${\displaystyle 1}$ ${\displaystyle p}$ ${\displaystyle \sigma ^{*}(p^{2})=1+p}$ ${\displaystyle n=p+1}$ ${\displaystyle k=p^{2}}$ ${\displaystyle n}$ ${\displaystyle \Box }$
• Let be an odd prime number and a number that is 3 larger than a prime number. Then the following applies: is not an untouchable number. Proof: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle n=p+3}$
  ${\displaystyle n}$
  
  • Let be an odd prime number. Then only has the real divisors and . Thus applies to the real partial sum . So there can never be an untouchable number because there is a number whose real prime is the same . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle 2p}$ ${\displaystyle 1,2}$ ${\displaystyle p}$ ${\displaystyle \sigma ^{*}(2p)=1+2+p=p+3}$ ${\displaystyle n=p+3}$ ${\displaystyle k=2p}$ ${\displaystyle n}$ ${\displaystyle \Box }$
• There are an infinite number of untouchable numbers. Their asymptotic density is at least evidence: by Paul Erdős and by Chen & Zhao ${\displaystyle d>0{,}06}$
  

Unsolved problems

• It is believed that the number is the only odd untouchable number. ${\displaystyle n=5}$
  • This would follow from Goldbach's strong conjecture if it were proven:
    A number (with prime numbers , ) only has the real divisors and . Thus the sum of their real factors is . If every even number can be represented as the sum of two different prime numbers (this is exactly the statement of the strong Goldbach hypothesis), then every odd number is not an untouchable number because it is the divisional sum of the number (with the real divisors and ). Furthermore is , and . Thus there can only be an odd untouchable number. ${\displaystyle k=p\cdot q}$ ${\displaystyle p,q\in \mathbb {P} }$ ${\displaystyle p\not =q}$ ${\displaystyle 1,p}$ ${\displaystyle q}$ ${\displaystyle \sigma ^{*}(k)=1+p+q}$ ${\displaystyle n>6}$ ${\displaystyle p+q}$ ${\displaystyle 1+p+q>7}$ ${\displaystyle k=p\cdot q}$ ${\displaystyle 1,p}$ ${\displaystyle q}$ ${\displaystyle 1=\sigma ^{*}(2)=1}$ ${\displaystyle 3=\sigma ^{*}(4)=1+2}$ ${\displaystyle 7=\sigma ^{*}(8)=1+2+4}$ ${\displaystyle n=5}$ ${\displaystyle \Box }$
• It is believed that all intangible numbers except 2 and 5 are composite numbers .
  • This would follow directly from the above assertion, especially since it states that apart from the number 5, only even numbers can be intangible numbers. Even numbers that are not equal to 2 are always composed. ${\displaystyle \Box }$

Web links

• Eric W. Weisstein : Untouchable Number . In: MathWorld (English).

Individual evidence

1. ↑ J. Sesiano: Two problems of number theory in Islamic times . In: Archive for History of Exact Sciences . tape 41 (3) , 1991, pp. 235–238 ( springer.com [accessed November 24, 2018]).
2. ^ Paul Erdős : About the numbers of the form and ${\displaystyle \sigma (n)-n}$ ${\displaystyle n-\phi (n)}$ . In: Elements of the Math. Band 28 , 1973, p. 83–86 ( emis.de [accessed November 24, 2018]).
3. ↑ Yong-Gao Chen , Qing-Qing Zhao : Nonaliquot numbers . In: Publicationes Mathematicae . tape 78 (2) , February 2011, p. 439-442 ( researchgate.net [accessed November 24, 2018]).
4. ↑ Eric W. Weisstein : Untouchable Number . In: MathWorld (English).