費馬數是以數學家費馬命名的一組自然數,具有形式:

其中為非負整數。

素數,可以得到n必須是2的。(若,其中b為奇數,則,即的因數。)也就是說,所有具有形式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五個。

基本性質

費馬數滿足以下的遞歸關係

 
 
 
 

其中 。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   有一個公因子 。那麼 能把

 

 都整除;則 能整除它們相減的差。因為 ,這使得 。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:

  •  的位數 可以表示成以 基數就是
  (參見高斯函數).
  • 除了 以外沒有費馬數可以表示成兩個素數的和。
  •  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  ,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[1],目前最小不確定質性的費馬數是 

不確定質性的數: ,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 的分解式:

 

歐拉證明費馬數的因數皆可表成 ,之後盧卡斯證明費馬數的因數皆可表成 

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

素性檢驗

方法一

 為第 個費馬數。如果 不等於零,那麼:

 是素數,當且僅當 

證明

假設以下等式成立:

 

那麼 ,因此滿足 的最小整數 一定整除 ,它是2的冪。另一方面, 不能整除 ,因此它一定等於 。特別地,存在至少 個小於 且與 互素的數,這只能在 是素數時才能發生。

假設 是素數。根據歐拉準則,有:

 ,

其中 勒讓德符號。利用重複平方,我們可以發現 ,因此 ,以及 。因為 ,根據二次互反律,我們便可以得出結論 

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 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:    

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