查看“︁皮萨诺周期”︁的源代码

{{NoteTA
|G1 = Math
}}
在[[数论]]中，自然数 ''n'' 的'''皮萨诺周期'''（通常记为π(''n'')）是[[斐波那契数列]]模 ''n'' 后的[[周期]]，以意大利数学家莱昂纳多·皮萨诺（即[[斐波那契]]）的名字命名。斐波那契数列取模後，周期的存在性曾在1774年为[[约瑟夫·拉格朗日]]所提及。<ref name="mathworld"><span class="citation mathworld" id="Reference-Mathworld-Pisano Period" contenteditable="false">[[埃里克·韦斯坦因|Weisstein, Eric W.]], [http://mathworld.wolfram.com/PisanoPeriod.html "Pisano Period"] {{Wayback|url=http://mathworld.wolfram.com/PisanoPeriod.html |date=20210225041010 }}, ''[[MathWorld]]''.</span></ref><ref>[http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf On Arithmetical functions related to the Fibonacci numbers] {{Wayback|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf |date=20201112034230 }}.</ref>

== 定义 ==
[[斐波那契数列]]是：
: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ... {{OEIS|A000045}}

斐波那契数列由下方的[[递推关系]]定义
: <math>F_0 = 0</math>
: <math>F_1 = 1</math>
: <math>F_i = F_{i-1} + F_{i-2}.</math>
对于任意整数''n'', 数列{''F<sub>i</sub>'' (mod ''n'')}为周期数列。皮萨诺周期''{{Pi}}''(''n'')记为该数列的周期。例如，模3的斐波那契数列前若干项为：
: 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... {{OEIS|id=A082115}}
这一数列以8为周期，故''{{Pi}}''(3) = 8.

== 性质 ==
除去''{{Pi}}''(2) =&nbsp;3 以外，皮萨诺周期必为偶数。这一性质的一个简单证明可由如下事实导出：

考虑斐波那契矩阵
: <math>
\mathbf F = \begin{bmatrix} 1 & 1\\1 & 0 \end{bmatrix} 
</math>
则π(n)应等同于矩阵 '''F '''在一般线性群''GL''<sub>2</sub>(ℤ<sub>''n''</sub>)的阶，其中''GL''<sub>2</sub>(ℤ<sub>''n''</sub>)表示在整数模 ''n ''环上全体二阶可逆矩阵构成的乘法群。由于'''F'''的行列式为−1, 可知在ℤ<sub>n</sub>中有(−1)<sup>''{{Pi}}''(n)</sup>&nbsp;=&nbsp;1, 故''{{Pi}}''(n)为偶数。<ref>[http://www.theoremoftheday.org/Binomial/PeriodicFib/TotDPeriodic.pdf A Theorem on Modular Fibonacci Periodicity] {{Wayback|url=http://www.theoremoftheday.org/Binomial/PeriodicFib/TotDPeriodic.pdf |date=20191112023310 }}.</ref>

当 ''m,n'' 互质时，由[[中国剩余定理]]即知''{{Pi}}''(''mn'')等于''{{Pi}}''(''m'')和''{{Pi}}''(''n'')的[[最小公倍数]]。例如，''{{Pi}}''(3) = 8 而''{{Pi}}''(4) = 6，由此可得''{{Pi}}''(12) = 24. 因此，对皮萨诺周期的研究可以化归为对[[素数幂]]''q'' = ''p''<sup>''k ''</sup> (''k'' ≥ 1) 的皮萨诺周期的研究。

可以证明，若''p''为素数，则''{{Pi}}''(''p''<sup>''k''</sup>)整除''p''<sup>''k''–1</sup>''{{Pi}}''(''p''). 有猜想认为<math>
\pi(p^k) = p^{k-1}\pi(p)
</math>对一切素数''p''及整数''k'' > 1成立。任何不满足该猜想的素数''p''都必然是一个[[沃尔-孙-孙素数]]，而这种素数被猜想并不存在。

因此对皮萨诺周期的研究可以被进一步化归为对素数的皮萨诺周期的研究。出于这种考虑，需要特别指出两个反常的素数。素数2的皮萨诺周期为奇数，而素数5的皮萨诺周期和其他素数相比“大得多”。这两个素数的幂的皮萨诺周期为：

* 若 ''n'' = 2<sup>''k''</sup>, 則 ''{{Pi}}''(''n'') = 3·2<sup>''k''–1</sup> = 3''n''<span class="visualhide">/</span>2<span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"></span>.
* 若 ''n'' = 5<sup>''k''</sup>, 則 ''{{Pi}}''(''n'') = 4·5<sup>''k''</sup> = 4''n''.
由此可知对''n'' = 2·5<sup>''k''</sup>有''{{Pi}}''(''n'') = 6''n''.

2和5以外的所有素数均属于共轭类<math>p \equiv \pm 1 \pmod {10}</math> 或 <math>p \equiv \pm 2 \pmod {5}</math>. 在这一情况下，在模p下由[[斐波那契数列|比内]]公式的类比可知''{{Pi}}''(''p'') 是 <span class="texhtml ">''x''<sup>2</sup> – ''x'' – 1 </span>的根模''p''的指数。当<math>p \equiv \pm 1 \pmod {10}</math>时，根据二次互反律，这两个根在 <math>\mathbb{F}_{p} = \mathbb{Z}/p\mathbb{Z}</math> 中，由[[费马小定理]]可知''{{Pi}}''(''p'')整除''p'' – 1. 例如，''{{Pi}}''(11) = 11 – 1 = 10，''{{Pi}}''(29) = (29 – 1)/2 = 14.

若<math>p \equiv \pm 2 \pmod {5},</math> 根据二次互反律，<span class="texhtml ">''x''<sup>2</sup> – ''x'' – 1</span> 的根不在<math>\mathbb{F}_{p}</math>内，而是在有限域
: <math>
\mathbb{F}_{p}[x]/(x^2 - x - 1)
</math>
中。注意到[[弗罗贝尼乌斯自同态]] <math>x \mapsto x^p</math> 将方程的两根''r''和''s''交换，因而''r''<sup>''p''</sup> = ''s，''故''r''<sup>''p''+1</sup> = –1. 由此可得''r''<sup>2(''p''+1)</sup> = 1, 故''r''的阶，也即''π(p)''，是2(''p''+1)除以某个奇数的商，因而必为4的倍数。在这种情况中，最小的三个满足 ''{{Pi}}''(''p'')''<2(p+1) ''的例子为''{{Pi}}''(47) = 2(47 + 1)/3 = 32, ''{{Pi}}''(107) = 2(107 + 1)/3 = 72 及''{{Pi}}''(113) = 2(113 + 1)/3 = 76.

据上述讨论，若''n'' = ''p''<sup>''k''</sup>是一个奇素数幂，满足''π''(''n'') > ''n'', 则 ''{{Pi}}''(''n'')/4 是一个不大于''n''的整数。利用皮萨诺周期的乘积性质，可得
: ''{{Pi}}''(''n'') &#x2264; 6''n'',
等号成立当且仅当''n''&nbsp;=&nbsp;2&nbsp;·&nbsp;5<sup>''r''</sup>, ''r''&nbsp;≥&nbsp;1. 最小的两个等号成立的例子为 ''{{Pi}}''(10)&nbsp;=&nbsp;60 及 ''{{Pi}}''(50)&nbsp;=&nbsp;300. 若 ''n'' 不能表示为 2&nbsp;·&nbsp;5<sup>''r ''</sup> 的形式，则''{{Pi}}''(''n'') &#x2264; 4''n''.

== 列表 ==
前十二个自然数的皮萨诺周期（[[整數數列線上大全|OEIS]]中的数列[[oeis:A001175|A001175]]）及其对应的一个周期内的所有数列举如下（为可读性起见，在每个0前加有空格；X, E分别表示10, 11）：<ref>[[oeis:A001175/a001175.jpg|Graph of the cycles modulo 1 to 24]]. </ref>
{| class="wikitable"
!''n''
!π(''n'')
!一个周期中0的数目 ({{Oeis|id=A001176}})
!一个周期中的所有数({{Oeis|id=A161553}})
![[整數數列線上大全|OEIS]] 中
|-
|1
|[[1]]
|1
|0
|{{OEIS link|id=A000004}}
|-
|2
|[[3]]
|1
|011
|{{OEIS link|id=A011655}}
|-
|3
|[[8]]
|2
|0112 0221
|{{OEIS link|id=A082115}}
|-
|4
|[[6]]
|1
|011231
|{{OEIS link|id=A079343}}
|-
|5
|[[20]]
|4
|01123 03314 04432 02241
|{{OEIS link|id=A082116}}
|-
|6
|[[24]]
|2
|011235213415 055431453251
|{{OEIS link|id=A082117}}
|-
|7
|[[16]]
|2
|01123516 06654261
|{{OEIS link|id=A105870}}
|-
|8
|[[12]]
|2
|011235 055271
|{{OEIS link|id=A079344}}
|-
|9
|[[24]]
|2
|011235843718 088764156281
|{{OEIS link|id=A007887}}
|-
|10
|[[60]]
|4
|011235831459437 077415617853819 099875279651673 033695493257291
|{{OEIS link|id=A003893}}
|-
|11
|[[10]]
|1
|01123582X1
|{{OEIS link|id=A105955}}
|-
|12
|[[24]]
|2
|011235819X75 055X314592E1
|{{OEIS link|id=A089911}}
|}
{| class="wikitable"
!π(''n'')
!+1
!+2
!+3
!+4
!+5
!+6
!+7
!+8
!+9
!+10
!+11
!+12
|-
!0+
|1
|3
|8
|6
|20
|24
|16
|12
|24
|60
|10
|24
|-
!12+
|28
|48
|40
|24
|36
|24
|18
|60
|16
|30
|48
|24
|-
!24+
|100
|84
|72
|48
|14
|120
|30
|48
|40
|36
|80
|24
|-
!36+
|76
|18
|56
|60
|40
|48
|88
|30
|120
|48
|32
|24
|-
!48+
|112
|300
|72
|84
|108
|72
|20
|48
|72
|42
|58
|120
|-
!60+
|60
|30
|48
|96
|140
|120
|136
|36
|48
|240
|70
|24
|-
!72+
|148
|228
|200
|18
|80
|168
|78
|120
|216
|120
|168
|48
|-
!84+
|180
|264
|56
|60
|44
|120
|112
|48
|120
|96
|180
|48
|-
!96+
|196
|336
|120
|300
|50
|72
|208
|84
|80
|108
|72
|72
|-
!108+
|108
|60
|152
|48
|76
|72
|240
|42
|168
|174
|144
|120
|-
!120+
|110
|60
|40
|30
|500
|48
|256
|192
|88
|420
|130
|120
|-
!132+
|144
|408
|360
|36
|276
|48
|46
|240
|32
|210
|140
|24
|}

== 斐波那契数的皮萨诺周期 ==
如果''n'' = ''F'' (2''k'') (''k'' ≥ 2), 那么π(''n'') = 4''k''; 如果''n'' = ''F'' (2''k'' + 1) (''k'' ≥ 2), 那么π(''n'') = 8''k'' + 4. 换而言之，模 F(2k) (k ≥ 2)的一个周期内有两个0，而模F (2k + 1) (k ≥ 2)的一个周期内有四个0.
{| class="wikitable"
!''k''
!''F'' (''k'')
!π(''F'' (''k''))
!截至首个0出现前的一个（对 k ≤ 3）或一半(对不小于4的偶数k)、四分之一个（对不小于4的奇数k）周期<br>
|-
|1
|1
|1
|0
|-
|2
|1
|1
|0
|-
|3
|2
|3
|0, 1, 1
|-
|4
|3
|8
|0, 1, 1, 2
|-
|5
|5
|20
|0, 1, 1, 2, 3
|-
|6
|8
|12
|0, 1, 1, 2, 3, 5
|-
|7
|13
|28
|0, 1, 1, 2, 3, 5, 8
|-
|8
|21
|16
|0, 1, 1, 2, 3, 5, 8, 13
|-
|9
|34
|36
|0, 1, 1, 2, 3, 5, 8, 13, 21
|-
|10
|55
|20
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34
|-
|11
|89
|44
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
|-
|12
|144
|24
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
|-
|13
|233
|52
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
|-
|14
|377
|28
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
|-
|15
|610
|60
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377
|-
|16
|987
|32
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
|-
|17
|1597
|68
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987
|-
|18
|2584
|36
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597
|-
|19
|4181
|76
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584
|-
|20
|6765
|40
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181
|-
|21
|10946
|84
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765
|-
|22
|17711
|44
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946
|-
|23
|28657
|92
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711
|-
|24
|46368
|48
|0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657
|}

== 參考文獻 ==
* <cite class="citation journal" contenteditable="false">Engstrom, H. T. (1931). "On sequences defined by linear recurrence relations". ''Trans. Am. Math. Soc.'' '''33''' (1): 210–218. [[DOI|doi]]:[[doi:10.1090/S0002-9947-1931-1501585-5|10.1090/S0002-9947-1931-1501585-5]]. [[JSTOR]]&nbsp;[//www.jstor.org/stable/1989467 1989467]. </cite><cite class="citation journal" contenteditable="false">[[數學評論|MR]]&nbsp;[//www.ams.org/mathscinet-getitem?mr=1501585 1501585].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=On+sequences+defined+by+linear+recurrence+relations&rft.aufirst=H.+T.&rft.aulast=Engstrom&rft.date=1931&rft.genre=article&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1501585&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F1989467&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1931-1501585-5&rft.issue=1&rft.jtitle=Trans.+Am.+Math.+Soc.&rft.pages=210-218&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=33" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Freyd, Peter; Brown, Kevin S. (1992). "Problems and Solutions: Solutions: E3410". ''Amer. Math. ''</cite><cite class="citation journal" contenteditable="false">''Monthly'' '''99''' (3): 278–279. [[DOI|doi]]:[[doi:10.2307/2325076|10.2307/2325076]].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=Problems+and+Solutions%3A+Solutions%3A+E3410&rft.au=Brown%2C+Kevin+S.&rft.aufirst=Peter&rft.aulast=Freyd&rft.date=1992&rft.genre=article&rft_id=info%3Adoi%2F10.2307%2F2325076&rft.issue=3&rft.jtitle=Amer.+Math.+Monthly&rft.pages=278-279&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=99" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Ward, Morgan (1931). "The characteristic number of a sequence of integers satisfying a linear recursion relation". ''Trans. Am. Math. ''</cite><cite class="citation journal" contenteditable="false">''Soc.'' '''33''' (1): 153–165. [[DOI|doi]]:[[doi:10.1090/S0002-9947-1931-1501582-X|10.1090/S0002-9947-1931-1501582-X]]. [[JSTOR]]&nbsp;[//www.jstor.org/stable/1989464 1989464].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=The+characteristic+number+of+a+sequence+of+integers+satisfying+a+linear+recursion+relation&rft.aufirst=Morgan&rft.aulast=Ward&rft.date=1931&rft.genre=article&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F1989464&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1931-1501582-X&rft.issue=1&rft.jtitle=Trans.+Am.+Math.+Soc.&rft.pages=153-165&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=33" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Ward, Morgan (1933). "The arithmetical theory of linear recurring series". ''Trans. Am. Math. Soc.'' '''35''' (3): 600–628. [[DOI|doi]]:[[doi:10.1090/S0002-9947-1933-1501705-4|10.1090/S0002-9947-1933-1501705-4]]. </cite><cite class="citation journal" contenteditable="false">[[JSTOR]]&nbsp;[//www.jstor.org/stable/1989851 1989851].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=The+arithmetical+theory+of+linear+recurring+series&rft.aufirst=Morgan&rft.aulast=Ward&rft.date=1933&rft.genre=article&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F1989851&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1933-1501705-4&rft.issue=3&rft.jtitle=Trans.+Am.+Math.+Soc.&rft.pages=600-628&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=35" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Zierler, Neal (1959). "Linear recurring sequences". ''J. SIAM'' '''7''' (1): 31–38. [[DOI|doi]]:[[doi:10.1137/0107003|10.1137/0107003]]. [[JSTOR]]&nbsp;[//www.jstor.org/stable/2099002 2099002]. </cite><cite class="citation journal" contenteditable="false">[[數學評論|MR]]&nbsp;[//www.ams.org/mathscinet-getitem?mr=0101979 0101979].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=Linear+recurring+sequences&rft.aufirst=Neal&rft.aulast=Zierler&rft.date=1959&rft.genre=article&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0101979&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2099002&rft_id=info%3Adoi%2F10.1137%2F0107003&rft.issue=1&rft.jtitle=J.+SIAM&rft.pages=31-38&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=7" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Wall, D. D. (1960). </cite><cite class="citation journal" contenteditable="false">"Fibonacci series modulo m". ''Am. Math. Monthly'' '''67''' (6): 525&#x2014;532. [[DOI|doi]]:[[doi:10.2307/2309169|10.2307/2309169]]. </cite><cite class="citation journal" contenteditable="false">[[JSTOR]]&nbsp;[//www.jstor.org/stable/2309169 2309169].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=Fibonacci+series+modulo+m&rft.aufirst=D.+D.&rft.aulast=Wall&rft.date=1960&rft.genre=article&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2309169&rft_id=info%3Adoi%2F10.2307%2F2309169&rft.issue=6&rft.jtitle=Am.+Math.+Monthly&rft.pages=525-532&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=67" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Bloom, D. M. (1965). "Periodicity in generalized Fibonacci sequences". ''Am. Math. Monthly'' '''72''': 856&#x2014;861. [[DOI|doi]]:[[doi:10.2307/2315029|10.2307/2315029]]. [[JSTOR]]&nbsp;[//www.jstor.org/stable/2315029 2315029]. </cite><cite class="citation journal" contenteditable="false">[[數學評論|MR]]&nbsp;[//www.ams.org/mathscinet-getitem?mr=0222015 0222015].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=Periodicity+in+generalized+Fibonacci+sequences&rft.aufirst=D.+M.&rft.aulast=Bloom&rft.date=1965&rft.genre=article&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0222015&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2315029&rft_id=info%3Adoi%2F10.2307%2F2315029&rft.jtitle=Am.+Math.+Monthly&rft.pages=856-861&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=72" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Laxton, R. R. (1969). "On groups of linear recurrences". ''Duke Mathematical Journal'' '''36''' (4): 721–736. [[DOI|doi]]:[[doi:10.1215/S0012-7094-69-03687-4|10.1215/S0012-7094-69-03687-4]]. </cite><cite class="citation journal" contenteditable="false">[[數學評論|MR]]&nbsp;[//www.ams.org/mathscinet-getitem?mr=0258781 0258781].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=On+groups+of+linear+recurrences&rft.aufirst=R.+R.&rft.aulast=Laxton&rft.date=1969&rft.genre=article&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0258781&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-69-03687-4&rft.issue=4&rft.jtitle=Duke+Mathematical+Journal&rft.pages=721-736&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=36" contenteditable="false">&nbsp;</span>
* <cite class="citation journal" contenteditable="false">Brent, Richard P. (1994). "On the periods of generalized Fibonacci sequences". ''Mathematics of Computation'' '''63''' (207): 389–401. [[arXiv]]:[//arxiv.org/abs/1004.5439 1004.5439]. [[Bibcode]]:[http://adsabs.harvard.edu/abs/1994MaCom..63..389B 1994MaCom..63..389B] {{Wayback|url=http://adsabs.harvard.edu/abs/1994MaCom..63..389B |date=20181116073147 }}. [[DOI|doi]]:[[doi:10.2307/2153583|10.2307/2153583]]. [[JSTOR]]&nbsp;[//www.jstor.org/stable/2153583 2153583]. </cite><cite class="citation journal" contenteditable="false">[[數學評論|MR]]&nbsp;[//www.ams.org/mathscinet-getitem?mr=1216256 1216256].</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.atitle=On+the+periods+of+generalized+Fibonacci+sequences&rft.aufirst=Richard+P.&rft.aulast=Brent&rft.date=1994&rft.genre=article&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1216256&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2153583&rft_id=info%3Aarxiv%2F1004.5439&rft_id=info%3Abibcode%2F1994MaCom..63..389B&rft_id=info%3Adoi%2F10.2307%2F2153583&rft.issue=207&rft.jtitle=Mathematics+of+Computation&rft.pages=389-401&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=63" contenteditable="false">&nbsp;</span>
* <cite class="citation web" contenteditable="false">Johnson, R. C. (2008). </cite><cite class="citation web" contenteditable="false">[http://maths.dur.ac.uk/~dma0rcj/PED/fib.pdf "Fibonacci numbers and matrices"] {{Wayback|url=http://maths.dur.ac.uk/~dma0rcj/PED/fib.pdf |date=20201130211206 }} (PDF).</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3APisano+period&rft.aufirst=R.+C.&rft.aulast=Johnson&rft.btitle=Fibonacci+numbers+and+matrices&rft.date=2008&rft.genre=unknown&rft_id=http%3A%2F%2Fmaths.dur.ac.uk%2F~dma0rcj%2FPED%2Ffib.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" contenteditable="false">&nbsp;</span>{{Reflist}}

[[Category:数论]]
[[Category:斐波那契数]]