全循环质数
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在数论中，全循环质数[1]:166又名长质数是指一个质数p，使分数1/p的循环节长度比质数少1，更精确地说，全循环质数是指一个质数p，在一个已知底数为b的进位制下，在下面算式中可以得出一个循环数的质数
若p为23，b为17，所得的数字0C9A5F8ED52G476B1823BE为循环数
- 0C9A5F8ED52G476B1823BE × 1 = 0C9A5F8ED52G476B1823BE
- 0C9A5F8ED52G476B1823BE × 2 = 1823BE0C9A5F8ED52G476B
- 0C9A5F8ED52G476B1823BE × 3 = 23BE0C9A5F8ED52G476B18
- 0C9A5F8ED52G476B1823BE × 4 = 2G476B1823BE0C9A5F8ED5
- 0C9A5F8ED52G476B1823BE × 5 = 3BE0C9A5F8ED52G476B182
- 0C9A5F8ED52G476B1823BE × 6 = 476B1823BE0C9A5F8ED52G
- 0C9A5F8ED52G476B1823BE × 7 = 52G476B1823BE0C9A5F8ED
- 0C9A5F8ED52G476B1823BE × 8 = 5F8ED52G476B1823BE0C9A
- 0C9A5F8ED52G476B1823BE × 9 = 6B1823BE0C9A5F8ED52G47
- 0C9A5F8ED52G476B1823BE × A = 76B1823BE0C9A5F8ED52G4
- 0C9A5F8ED52G476B1823BE × B = 823BE0C9A5F8ED52G476B1
- 0C9A5F8ED52G476B1823BE × C = 8ED52G476B1823BE0C9A5F
- 0C9A5F8ED52G476B1823BE × D = 9A5F8ED52G476B1823BE0C
- 0C9A5F8ED52G476B1823BE × E = A5F8ED52G476B1823BE0C9
- 0C9A5F8ED52G476B1823BE × F = B1823BE0C9A5F8ED52G476
- 0C9A5F8ED52G476B1823BE × G = BE0C9A5F8ED52G476B1823
- 0C9A5F8ED52G476B1823BE × 10 = C9A5F8ED52G476B1823BE0
- 0C9A5F8ED52G476B1823BE × 11 = D52G476B1823BE0C9A5F8E
- 0C9A5F8ED52G476B1823BE × 12 = E0C9A5F8ED52G476B1823B
- 0C9A5F8ED52G476B1823BE × 13 = ED52G476B1823BE0C9A5F8
- 0C9A5F8ED52G476B1823BE × 14 = F8ED52G476B1823BE0C9A5
- 0C9A5F8ED52G476B1823BE × 15 = G476B1823BE0C9A5F8ED52
而，循环节长度为22，比23少1，因此23为全循环质数
十进制中的全循环质数有：
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593,... （OEIS数列A001913）
参见
参考文献
- ^ Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.
- Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers"; in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240–246.