全循環質數
在數論中，全循環質數[1]:166又名長質數是指一個質數p，使分數1/p的循環節長度比質數少1，更精確地說，全循環質數是指一個質數p，在一個已知底數為b的進位制下，在下面算式中可以得出一個循環數的質數
若p為23，b為17，所得的數字0C9A5F8ED52G476B1823BE為循環數
- 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，循環節長度為22，比23少1，因此23為全循環質數
十進位中的全循環質數有：
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數列A001913）
參考文獻
- 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.
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