Fermat Number - Factorization of Fermat Numbers

Factorization of Fermat Numbers

Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project Fermatsearch has successfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving the above mentioned result by Euler, proved in 1878 that every factor of Fermat number, with n at least 2, is of the form (see Proth number), where k is a positive integer; this is in itself almost sufficient to prove the primality of the known Fermat primes.

Factorizations of the first twelve Fermat numbers are:

F₀	=	21	+	1	=	3 is prime
F₁	=	22	+	1	=	5 is prime
F₂	=	24	+	1	=	17 is prime
F₃	=	28	+	1	=	257 is prime
F₄	=	216	+	1	=	65,537 is the largest known Fermat prime
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

As of February 2012, only F₀ to F₁₁ have been completely factored. The distributed computing project Fermat Search is searching for new factors of Fermat numbers. The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.

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