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.
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![i ^ {{mn}} \ equiv 1 {\ pmod {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789e9c0c223929b9145a5cfcdb13de9190c57b81)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![K_n](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2b988ea630d2c5571afe47efa3d3b251708acb)
The special cases are the Carmichael numbers .
![K_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa)
Each composite number is a dumpling number by betting. With which is Euler phi function meant.
![{\ displaystyle n: = m- \ varphi (m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2804ddf4e966751fb29d29a873a033c13e118f36)
![\ varphi (m)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb43fceeba71e55e9827d94cf51d8effdf46584)
Examples
Example 1:
Be and![{\ displaystyle n: = 4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e32c943ed163175138afb2ec7c028393cee78d5d)
Then the numbers and are too coprime. The following applies:
![{\ displaystyle i = 1,5,7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9f2033d33663e7abd021142a921a96be873a78)
![11](https://wikimedia.org/api/rest_v1/media/math/render/svg/da6aabe7c6af49fe640b2d401cb2dbe909bb7475)
![{\ displaystyle m = 12}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bb0dfab936fb6e6e8dfe729b2d48854c5b614b)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b70a322768b3815e7c4d3762d915d153a00de6)
Thus, all of the relatively prime numbers satisfy the congruence .
![{\ displaystyle m = 12}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bb0dfab936fb6e6e8dfe729b2d48854c5b614b)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![i ^ {{mn}} \ equiv 1 {\ pmod {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789e9c0c223929b9145a5cfcdb13de9190c57b81)
So there is a dumpling number for the number 4 and you write .
![{\ displaystyle m = 12}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bb0dfab936fb6e6e8dfe729b2d48854c5b614b)
![{\ displaystyle 12 \ in K_ {4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426edb299403b17d9bccfb22ac5a4c9ce66a67c6)
Example 2:
Be and![{\ displaystyle n: = 4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e32c943ed163175138afb2ec7c028393cee78d5d)
Then the numbers and are too coprime. The following applies:
![{\ displaystyle i = 1,3,5,9,11}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01b50f0a981cec7180c091b177e64ef14bcbfe22)
![13](https://wikimedia.org/api/rest_v1/media/math/render/svg/d478c234d544278fb494e9610b7b3310567302b0)
![{\ displaystyle m = 14}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0326db691a575525207191bb00b9602d022f22)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74e6a3b881744dcbd80d71b2e84841f75f064b50)
Thus, not all of the relatively prime numbers satisfy the congruence .
![{\ displaystyle m = 14}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0326db691a575525207191bb00b9602d022f22)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![i ^ {{mn}} \ equiv 1 {\ pmod {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789e9c0c223929b9145a5cfcdb13de9190c57b81)
Actually, you could have settled the calculation at . So there is no dumpling number for the number 4 and you write .
![i = 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9aea65b32ee831db79412d978e090135c02e54c)
![{\ displaystyle m = 14}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0326db691a575525207191bb00b9602d022f22)
![{\ displaystyle 14 \ not \ in K_ {4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a33ade87464dada642a8f18e355f34dc478674)
Example 3:
The following is a list of the first elements of the sets to :
![K_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa)
![{\ displaystyle K_ {10}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/606190885ecc76969d297ba4574b405c82eec7b4)
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, ...} |
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|
10 |
{12, 24, 28, 30, 50, 70, 110, 130, ...} |
Follow A208158 in OEIS
|
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