Practical number
In number theory , a practical number (from English practical number , also panarithmic number ) is a natural number with the property that every smaller number can be written as the sum of different real divisors of .
The mathematician AK Srinivasan first mentioned these numbers in 1948.
Examples
- The number has the proper factors and . All smaller numbers can be written as the sum of these real factors:
- Apparently one can write all numbers as the sum of these real divisors. So is a practical number.
- The number has the proper factors and . One can write the following smaller numbers as the sum of these real factors:
- 1 = 1, 2 = 2, 3 = 2 + 1, ...
- But even the number 4 can no longer be written as the sum of these real divisors, because the requirement sum of real divisors that differ in pairs would have to be broken, namely 4 = 2 + 2. So the number is not a practical number.
- The following numbers are the smallest practical numbers (which is 198th practical number ):
- 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252,…, 984, 990, 992, 1000, 1008,… ( continuation A005153 in OEIS )
properties
- A practical number is never a deficient number (so the sum of all real divisors of a practical number is never less than the number itself).
- All numbers in the form with are practical numbers.
- All even perfect numbers are practical numbers.
- Let be the product of powers (non-zero) of the first prime numbers. Then:
- is a practical number.
- In particular, every prime faculty (i.e. every product of the first prime numbers with ) is a practical number.
- All composite numbers are practical numbers.
- All numbers of the form with , i.e. every power of two , are practical numbers.
- The only odd practical number is .
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Proof:
- Suppose it is an odd number. Then that number has the proper factors and some major factors . However, this means that the number 2 cannot be represented as the sum of its real factors. Thus, it can not be a practical number.
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Proof:
- Be a practical number. Then:
- is a multiple of 4 or 6 (or both numbers).
- This statement has been proven by AK Srinivasan.
- The product of two practical numbers is again a practical number. The set of practical numbers is therefore closed with regard to multiplication.
- Proof: see
- Be a practical number and a divisor of . Then:
- is also a practical number.
- The least common multiple ( LCM for short ) of two practical numbers is again a practical number.
- There are infinitely many Fibonacci numbers , which are practical numbers. They are called practical Fibonacci numbers . The smallest are:
- For every positive rational number :
- There is a representation of as the sum of finitely many stem fractions with pairwise different denominators , where each denominator is a practical number (a so-called division into Egyptian fractions ).
- In other words: be . Then:
- with pairs of different practical numbers and
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Example 1:
- Be . Then you can use a special procedure to break this number down into the four original fractions
- This division is a division into Egyptian fractions. You get the denominators and . In fact, these four denominators are all practical numbers.
- Practical numbers are not always obtained immediately with this procedure, but there is always such a representation:
- Be . Then you can use a special procedure to break this number down into the four original fractions
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Example 2:
- Be . Then you can use the same procedure to break this number down into the four original fractions
- This division is not a division into Egyptian fractions. You get the denominators and . While three of these four denominators are practical numbers ( and ), it is not a practical number. So you have to force the fraction to the next best denominator, which is a practical number, namely . Ultimately, you get the following Egyptian fractions:
- You get the denominators and . In fact, these five denominators are all practical numbers.
- Be . Then you can use the same procedure to break this number down into the four original fractions
- Any even natural number can be represented as the sum of two practical numbers and .
- There are infinitely many triples (= three-tuples) of practical numbers.
- There is always at least one practical number in the interval for all positive real numbers .
- Let be the number of practical numbers that are less than or equal to. Then:
- with and a constant
Unsolved problems
- It is believed that there are infinitely many five-tuples of practical numbers.
Primitive practical numbers
A primitive practical number is a practical number that has one of the following two properties:
- It is square-free .
- If you divide it by one of its prime factors whose exponent is greater than , the result can no longer be a practical number.
Examples
- The number , as shown above , is a practical number. It is not square-free because it is. If you divide it by one of its prime factors whose exponent is greater than , that is, by , you get which is not a practical number. Thus is a primitive practical number.
- The number is a practical number. Your prime factorization is . If you divide it by one of its prime factors whose exponent is greater than , that is, by or by , you get or , respectively , which are both still practical numbers. Thus it is not a primitive practical number.
- The smallest primitive practical numbers are the following (there are exactly 61 of them, which are less than or equal to 1000):
- 1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888, 930, 966, 984, 1032, ... ( continuation A267124 in OEIS )
properties
- Every prime faculty (i.e. every product of the first prime numbers with ) is a primitive practical number.
Web links
- Eric W. Weisstein : Practical Number . In: MathWorld (English).
- Melfi, Practical numbers
Individual evidence
- ↑ a b c d e f g A. K. Srinivasan: Practical numbers. Current Science 17 , 1948, pp. 179-180 , accessed January 1, 2019 .
- ^ A b Ross Honsberger : Mathematical gems. Vieweg , 1981, p. 112 , accessed January 1, 2019 .
- ↑ Eric W. Weisstein : Practical Number . In: MathWorld (English).
- ↑ Maurice Margenstern : Les nombres pratiques: théorie, observations et conjectures. (PDF) Journal of Number Theory 37 , 1991, pp. 1–36 , accessed on January 1, 2019 .
- ^ Zhi-Wei Sun : A Conjecture on Unit Fractions Involving Primes. (PDF) Nanjing University , Department of Mathematics, 2015, pp. 1–15 , accessed January 1, 2019 .
- ^ David Eppstein : Egyptian fractions with practical denominators. 2016, accessed January 1, 2019 .
- ^ A b Giuseppe Melfi : On two conjecture about practical numbers. (PDF) Journal of Number Theory 56 (1), 1996, pp. 205–210 , accessed on January 1, 2019 .
- ^ A b c Giuseppe Melfi : A survey on practical numbers. (PDF) rend. Sem. Mat. Univ. Pole. Torino , 53 , 1995, pp. 347-359 , accessed January 1, 2019 .
- ↑ Miriam Hausman , Harold N. Shapiro : On Practical Numbers. Communications on Pure and Applied Mathematics , 37 (5), 1984, pp. 705-713 , accessed January 1, 2019 .
- ^ Andreas Weingartner : Practical numbers and the distribution of divisors. The Quarterly Journal of Mathematics , 66 (2), 2015, pp. 743-758 , accessed January 1, 2019 .