費馬數

費馬數是以數學家費馬命名的一組自然數，具有形式：

${\displaystyle F_{n}=2^{2^{n}}+1}$

其中${\displaystyle n}$為非負整數。

若${\displaystyle 2^{n}+1}$是素數，可以得到n必須是2的冪。（若${\displaystyle n=ab}$，其中${\displaystyle 1<a,b<n}$且b為奇數，則${\displaystyle 2^{n}+1\equiv (2^{a})^{b}+1\equiv (-1)^{b}+1\equiv 0{\pmod {2^{a}+1}}}$，即${\displaystyle 2^{a}+1}$是${\displaystyle 2^{n}+1}$的因數。）也就是說，所有具有形式${\displaystyle 2^{n}+1}$的素數必然是費馬數，這些素數稱為費馬素數。已知的費馬素數只有${\displaystyle F_{0}}$至${\displaystyle F_{4}}$五個。

基本性質

費馬數滿足以下的遞歸關係：

${\displaystyle F_{n}=(F_{n-1}-1)^{2}+1\,}$

${\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}$

${\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}$

${\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}$

其中${\displaystyle n\geqslant 2}$。這些等式都可以用數學歸納法推出。從最後一個等式中，我們可以推出哥德巴赫定理：任何兩個費馬數都沒有大於1的公因子。要推出這個，我們需要假設${\displaystyle 0\leqslant i<j}$且${\displaystyle F_{i}}$和${\displaystyle F_{j}}$有一個公因子${\displaystyle a>1}$。那麼${\displaystyle a}$能把

${\displaystyle F_{0}\cdots F_{j-1}}$

和${\displaystyle F_{j}}$都整除；則${\displaystyle a}$能整除它們相減的差。因為${\displaystyle a>1}$，這使得${\displaystyle a=2}$。造成矛盾。因為所有的費馬數顯然是奇數。作為一個推論，我們得到素數個數無窮的又一個證明。

其他性質：

• ${\displaystyle F_{n}}$的位數${\displaystyle D(n,b)}$可以表示成以${\displaystyle b}$為基數就是

${\displaystyle D(n,b)=\left\lfloor \log _{b}\left(2^{2^{\overset {n}{}}}+1\right)+1\right\rfloor \approx \lfloor 2^{n}\,\log _{b}2+1\rfloor }$ (參見高斯函數).

• 除了${\displaystyle F_{1}=2+3}$以外沒有費馬數可以表示成兩個素數的和。
• 當${\displaystyle p}$是奇素數的時候，沒有費馬數可以表示成兩個數的p次方相減的形式。
• 除了${\displaystyle F_{0}}$和${\displaystyle F_{1}}$，費馬數的最後一位是7。
• 所有費馬數（OEIS數列A051158）的倒數之和是無理數。 (所羅門·格倫布,1963)

費馬數的因式分解

最小的12個費馬數為：

F0	=	21	+	1	=	3
F1	=	22	+	1	=	5
F2	=	24	+	1	=	17
F3	=	28	+	1	=	257
F4	=	216	+	1	=	65,537 以上5個是已知的費馬素數。
F5	=	232	+	1	=	4,294,967,297
					=	641 × 6,700,417
F6	=	264	+	1	=	18,446,744,073,709,551,617
					=	274,177 × 67,280,421,310,721
F7	=	2128	+	1	=	340,282,366,920,938,463,463,374,607,431,768,211,457
					=	59,649,589,127,497,217 × 5,704,689,200,685,129,054,721
F8	=	2256	+	1	=	115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,937
					=	1,238,926,361,552,897 × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321
F9	=	2512	+	1	=	13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546, 976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,097
					=	2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 × 741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759,504,705,008,092,818,711,693,940,737
F10	=	21024	+	1	=	179,769,313,486,231,590,772,930,519,078,902,473,361,797,697,894,230,657,273,430,081,157,732,675,805,500,963,132,708,477,322,407,536,021,120, 113,879,871,393,357,658,789,768,814,416,622,492,847,430,639,474,124,377,767,893,424,865,485,276,302,219,601,246,094,119,453,082,952,085, 005,768,838,150,682,342,462,881,473,913,110,540,827,237,163,350,510,684,586,298,239,947,245,938,479,716,304,835,356,329,624,224,137,217
					=	45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 × 130,439,874,405,488,189,727,484,768,796,509,903,946,608,530,841,611,892,186,895,295,776,832,416,251,471,863,574, 140,227,977,573,104,895,898,783,928,842,923,844,831,149,032,913,798,729,088,601,617,946,094,119,449,010,595,906, 710,130,531,906,171,018,354,491,609,619,193,912,488,538,116,080,712,299,672,322,806,217,820,753,127,014,424,577
F11	=	22048	+	1	=	32,317,006,071,311,007,300,714,876,688,669,951,960,444,102,669,715,484,032,130,345,427,524,655,138,867,890,893,197,201,411,522,913,463,688,717, 960,921,898,019,494,119,559,150,490,921,095,088,152,386,448,283,120,630,877,367,300,996,091,750,197,750,389,652,106,796,057,638,384,067, 568,276,792,218,642,619,756,161,838,094,338,476,170,470,581,645,852,036,305,042,887,575,891,541,065,808,607,552,399,123,930,385,521,914, 333,389,668,342,420,684,974,786,564,569,494,856,176,035,326,322,058,077,805,659,331,026,192,708,460,314,150,258,592,864,177,116,725,943, 603,718,461,857,357,598,351,152,301,645,904,403,697,613,233,287,231,227,125,684,710,820,209,725,157,101,726,931,323,469,678,542,580,656, 697,935,045,997,268,352,998,638,215,525,166,389,437,335,543,602,135,433,229,604,645,318,478,604,952,148,193,555,853,611,059,596,230,657
					=	319,489 × 974,849 × 167,988,556,341,760,475,137 × 3,560,841,906,445,833,920,513 × 173,462,447,179,147,555,430,258,970,864,309,778,377,421,844,723,664,084,649,347,019,061,363,579,192,879,108,857,591,038,330,408,837,177,983,810,868,451, 546,421,940,712,978,306,134,189,864,280,826,014,542,758,708,589,243,873,685,563,973,118,948,869,399,158,545,506,611,147,420,216,132,557,017,260,564,139, 394,366,945,793,220,968,665,108,959,685,482,705,388,072,645,828,554,151,936,401,912,464,931,182,546,092,879,815,733,057,795,573,358,504,982,279,280,090, 942,872,567,591,518,912,118,622,751,714,319,229,788,100,979,251,036,035,496,917,279,912,663,527,358,783,236,647,193,154,777,091,427,745,377,038,294, 584,918,917,590,325,110,939,381,322,486,044,298,573,971,650,711,059,244,462,177,542,540,706,913,047,034,664,643,603,491,382,441,723,306,598,834,177

其中前八個來源於（OEIS數列A000215）。

質因數個數顏色： 紅色：2個因數；綠色：3個因數；粉色：4個因數；藍色：5個因數；橘色：6個因數；紫色：7個因數(含以上)；

只有最小的12個費馬數被人們完全分解了^[1]，目前最小不確定質性的費馬數是${\displaystyle F_{33}}$。

不確定質性的數：${\displaystyle F_{n}}$，n = 33、34、35、44、45、46、...

歷史

1640年，費馬提出了一個猜想，認為所有的費馬數都是素數。這一猜想對最小的5個費馬數成立，於是費馬宣稱他找到了表示素數的公式。然而，歐拉在1732年否定了這一猜想，他給出了${\displaystyle F_{5}}$的分解式：

${\displaystyle F_{5}=2^{32}+1=4294967297=641\times 6700417}$

歐拉證明費馬數的因數皆可表成${\displaystyle k2^{n+1}+1}$，之後盧卡斯證明費馬數的因數皆可表成${\displaystyle k2^{n+2}+1}$。

費馬的本猜測到了大數可說是大錯特錯，甚至多數數學家認為不存在第6個費馬質數。

定理

主條目：可作圖多邊形

• 高斯稱：尺規作圖正多邊形的邊數目的充分條件是2的非負整數次方乘以任意個（可為0個）不同的費馬素數的積，解決了兩千年來懸而未決的難題。這個條件也是必要條件，但他沒有給出證明。

素性檢驗

方法一

設${\displaystyle F_{n}=2^{2^{n}}+1}$為第${\displaystyle n}$個費馬數。如果${\displaystyle n}$不等於零，那麼：

${\displaystyle F_{n}}$是素數，當且僅當${\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}$。

證明

假設以下等式成立：

${\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}}$

那麼${\displaystyle 3^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}$，因此滿足${\displaystyle 3^{k}=1{\pmod {F_{n}}}}$的最小整數${\displaystyle k}$一定整除${\displaystyle F_{n}-1=2^{2^{n}}}$，它是2的冪。另一方面，${\displaystyle k}$不能整除${\displaystyle {\tfrac {F_{n}-1}{2}}}$，因此它一定等於${\displaystyle F_{n}-1}$。特別地，存在至少${\displaystyle F_{n}-1}$個小於${\displaystyle F_{n}}$且與${\displaystyle F_{n}}$互素的數，這只能在${\displaystyle F_{n}}$是素數時才能發生。

假設${\displaystyle F_{n}}$是素數。根據歐拉準則，有：

${\displaystyle 3^{\frac {F_{n}-1}{2}}\equiv \left({\frac {3}{F_{n}}}\right){\pmod {F_{n}}}}$,

其中${\displaystyle \left({\frac {3}{F_{n}}}\right)}$是勒讓德符號。利用重複平方，我們可以發現${\displaystyle 2^{2^{n}}\equiv 1{\pmod {3}}}$，因此${\displaystyle F_{n}\equiv 2{\pmod {3}}}$，以及${\displaystyle \left({\frac {F_{n}}{3}}\right)=-1}$。因為${\displaystyle F_{n}\equiv 1{\pmod {4}}}$，根據二次互反律，我們便可以得出結論${\displaystyle \left({\frac {3}{F_{n}}}\right)=-1}$。

方法二

根據費馬平方和定理，可證明某些費馬數為合數。(因為${\displaystyle 4n+1}$型的質數皆可表示為兩正整數的平方和，且表示方法唯一) 例如： ${\displaystyle F_{5}=(2^{16})^{2}+1^{2}=20449^{2}+62264^{2}}$ ${\displaystyle F_{6}=(2^{32})^{2}+1^{2}=1438793759^{2}[phone]^{2}}$

注釋

1. （英文） 費馬數的分解 （頁面存檔備份，存於網際網路檔案館）