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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

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

,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中的数列A001913

參見编辑

參考文獻编辑

  1. ^ Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
  1. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.
  2. 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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