Dumpling number

In number theory , a dumpling number for a given integer is a composite number with the property that all coprime values satisfy the congruence . This property is named after Walter Knödel . The set of all dumpling numbers from is denoted by. ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle i<m}$ ${\displaystyle i^{m-n}\equiv 1{\pmod {m}}}$ ${\displaystyle n}$ ${\displaystyle K_{n}}$

The special cases are the Carmichael numbers . ${\displaystyle K_{1}}$

Each composite number is a dumpling number by betting. With which is Euler phi function meant. ${\displaystyle n:=m-\varphi (m)}$ ${\displaystyle \varphi (m)}$

Examples

Example 1:

Be and ${\displaystyle n:=4}$ ${\displaystyle m:=12.}$

Then the numbers and are too coprime. The following applies: ${\displaystyle i=1,5,7}$ ${\displaystyle 11}$ ${\displaystyle m=12}$

${\displaystyle {\begin{aligned}1^{12-4}&=&1&&&\equiv &1{\pmod {12}}\\5^{12-4}&=&390625&=&32552\cdot 12+1&\equiv &1{\pmod {12}}\\7^{12-4}&=&5764801&=&480400\cdot 12+1&\equiv &1{\pmod {12}}\\11^{12-4}&=&214358881&=&17863240\cdot 12+1&\equiv &1{\pmod {12}}\end{aligned}}}$

Thus, all of the relatively prime numbers satisfy the congruence . ${\displaystyle m=12}$ ${\displaystyle i}$ ${\displaystyle i^{m-n}\equiv 1{\pmod {m}}}$

So there is a dumpling number for the number 4 and you write . ${\displaystyle m=12}$ ${\displaystyle 12\in K_{4}}$

Example 2:

Be and ${\displaystyle n:=4}$ ${\displaystyle m:=14.}$

Then the numbers and are too coprime. The following applies: ${\displaystyle i=1,3,5,9,11}$ ${\displaystyle 13}$ ${\displaystyle m=14}$

${\displaystyle {\begin{aligned}1^{14-4}&=&1&&&&\equiv &\quad 1&{\pmod {14}}\\3^{14-4}&=&59049&=&4217\cdot 14&+11&\equiv &\quad 11&{\pmod {14}}\\5^{14-4}&=&9765625&=&697544\cdot 14&+9&\equiv &\quad 9&{\pmod {14}}\\9^{14-4}&=&3486784401&=&249056028\cdot 14&+9&\equiv &\quad 9&{\pmod {14}}\\11^{14-4}&=&25937424601&=&1852673185\cdot 14&+11&\equiv &\quad 11&{\pmod {14}}\\13^{14-4}&=&137858491849&=&9847035132\cdot 14&+1&\equiv &\quad 1&{\pmod {14}}\end{aligned}}}$

Thus, not all of the relatively prime numbers satisfy the congruence . ${\displaystyle m=14}$ ${\displaystyle i}$ ${\displaystyle i^{m-n}\equiv 1{\pmod {m}}}$

Actually, you could have settled the calculation at . So there is no dumpling number for the number 4 and you write . ${\displaystyle i=3}$ ${\displaystyle m=14}$ ${\displaystyle 14\not \in K_{4}}$

Example 3:

The following is a list of the first elements of the sets to : ${\displaystyle K_{1}}$ ${\displaystyle K_{10}}$

n	K n
1	{561, 1105, 1729, 2465, 2821, 6601, ...}	Follow A002997 in OEIS
2	{4, 6, 8, 10, 12, 14, 22, 24, 26, ...}	Follow A050990 in OEIS
3	{9, 15, 21, 33, 39, 51, 57, 63, 69, ...}	Follow A033553 in OEIS
4th	{6, 8, 12, 16, 20, 24, 28, 40, 44, ...}	Follow A050992 in OEIS
5	{25, 65, 85, 145, 165, 185, 205, 265, ...}	Follow A050993 in OEIS
6th	{8, 10, 12, 18, 24, 30, 36, 42, 66, ...}	Follow A208154 in OEIS
7th	{15, 49, 91, 133, 217, 259, 301, 427, ...}	Follow A208155 in OEIS
8th	{12, 14, 16, 20, 24, 32, 40, 48, 56, ...}	Follow A208156 in OEIS
9	{21, 27, 45, 63, 99, 105, 117, 153, ...}	Follow A208157 in OEIS
10	{12, 24, 28, 30, 50, 70, 110, 130, ...}	Follow A208158 in OEIS

literature

• A. Makowski: Generalization of Morrow's D-Numbers 1963, p. 71.
• Paulo Ribenboim : The New Book of Prime Number Records . Springer-Verlag, New York 1989, ISBN 978-0-387-94457-9 , p. 101.