# 卡邁克爾數

## 概觀

Korselt雖然發現了這些性質，但不能找到例子。1910年羅伯特·丹尼·卡邁克爾找到了第一個兼最小的有這樣性質的數——561。$561 = 3 \times 11 \times 17$，無平方数因数，且2|560 ; 10|560 ; 16|560 。

1105 = 5×13×17 (4 | 1104, 12 | 1104, 16 | 1104)
1729 = 7×13×19 (6 | 1728, 12 | 1728, 18 | 1728)
2465 = 5×17×29 (4 | 2464, 16 | 2464, 28 | 2464)
2821 = 7×13×31 (6 | 2820, 12 | 2820, 30 | 2820)
6601 = 7×23×41 (6 | 6600, 22 | 6600, 40 | 6600)
8911 = 7×19×67 (6 | 8910, 18 | 8910, 66 | 8910)


J. Chernick 在1939年證明的一個定理，可以構造卡邁克爾數的一個子集

1992年Löh和Niebuhr找到一些很大的卡邁克爾數，其中一個有1 101 518 個因數且有多於$1.6 \times 10^7$個位數。

### 性質

k
3 561 = 3×11×17
4 41041 = 7×11×13×41
5 825265 = 5×7×17×19×73
6 321197185 = 5×19×23×29×37×137
7 5394826801 = 7×13×17×23×31×67×73
8 232250619601 = 7×11×13×17×31×37×73×163
9 9746347772161 = 7×11×13×17×19×31×37×41×641

i
1 41041 = 7×11×13×41
2 62745 = 3×5×47×89
3 63973 = 7×13×19×37
4 75361 = 11×13×17×31
5 101101 = 7×11×13×101
6 126217 = 7×13×19×73
7 172081 = 7×13×31×61
8 188461 = 7×13×19×109
9 278545 = 5×17×29×113
10 340561 = 13×17×23×67

## 參考

• Chernick, J. (1935). On Fermat's simple theorem. Bull. Amer. Math. Soc. 45, 269–274.
• Ribenboim, Paolo (1996). The New Book of Prime Number Records.
• Howe, Everett W. (2000). Higher-order Carmichael numbers. Mathematics of Computation 69, 1711–1719. (online version)
• Löh, Günter and Niebuhr, Wolfgang (1996). A new algorithm for constructing large Carmichael numbers(pdf)
• Korselt (1899). Probleme chinois. L'intermediaire des mathematiciens, 6, 142–143.
• Carmichael, R. D. (1912) On composite numbers P which satisfy the Fermat congruence $a^{P-1}\equiv 1\bmod P$. Am. Math. Month. 19 22–27.
• Erdős, Paul (1956). On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4, 201 –206.
• Alford, Granville and Pomerance (1994). There are infinitely many Carmichael numbers, Ann. of Math. 140(3), 703–722.