<P> The following is a list of the known perfect numbers, and the exponents p that can be used to generate them (using the expression 2 × (2 − 1)) whenever 2 − 1 is a Mersenne prime . All even perfect numbers are of this form . It is not known whether there are any odd perfect numbers . As of 2018 there are 50 known perfect numbers in total . The ratio p / digits approaches log (10) / log (4) = 1.6609640474...</P> <Table> <Tr> <Th> Rank </Th> <Th> p </Th> <Th> Perfect number </Th> <Th> Digits </Th> <Th> Year </Th> <Th> Discoverer </Th> </Tr> <Tr> <Td> </Td> <Td> </Td> <Td> 6 </Td> <Td> </Td> <Td> 4th century B.C. </Td> <Td> Euclid </Td> </Tr> <Tr> <Td> </Td> <Td> </Td> <Td> 28 </Td> <Td> </Td> <Td> 4th century B.C. </Td> <Td> Euclid </Td> </Tr> <Tr> <Td> </Td> <Td> 5 </Td> <Td> 496 </Td> <Td> </Td> <Td> 4th century B.C. </Td> <Td> Euclid </Td> </Tr> <Tr> <Td> </Td> <Td> 7 </Td> <Td> 8128 </Td> <Td> </Td> <Td> 4th century B.C. </Td> <Td> Euclid </Td> </Tr> <Tr> <Td> 5 </Td> <Td> 13 </Td> <Td> 33550336 </Td> <Td> 8 </Td> <Td> 1456 </Td> <Td> First seen in a medieval manuscript, Munich, Bayerische Staatsbibliothek, CLM 14908, fol. 33 </Td> </Tr> <Tr> <Td> 6 </Td> <Td> 17 </Td> <Td> 8589869056 </Td> <Td> 10 </Td> <Td> 1588 </Td> <Td> Cataldi </Td> </Tr> <Tr> <Td> 7 </Td> <Td> 19 </Td> <Td> 137438691328 </Td> <Td> 12 </Td> <Td> 1588 </Td> <Td> Cataldi </Td> </Tr> <Tr> <Td> 8 </Td> <Td> 31 </Td> <Td> 2305843008139952128 </Td> <Td> 19 </Td> <Td> 1772 </Td> <Td> Euler </Td> </Tr> <Tr> <Td> 9 </Td> <Td> 61 </Td> <Td> 265845599156...615953842176 </Td> <Td> 37 </Td> <Td> 1883 </Td> <Td> Pervushin </Td> </Tr> <Tr> <Td> 10 </Td> <Td> 89 </Td> <Td> 191561942608...321548169216 </Td> <Td> 54 </Td> <Td> 1911 </Td> <Td> Powers </Td> </Tr> <Tr> <Td> 11 </Td> <Td> 107 </Td> <Td> 131640364585...117783728128 </Td> <Td> 65 </Td> <Td> 1914 </Td> <Td> Powers </Td> </Tr> <Tr> <Td> 12 </Td> <Td> 127 </Td> <Td> 144740111546...131199152128 </Td> <Td> 77 </Td> <Td> 1876 </Td> <Td> Lucas </Td> </Tr> <Tr> <Td> 13 </Td> <Td> 521 </Td> <Td> 235627234572...160555646976 </Td> <Td> 314 </Td> <Td> 1952 </Td> <Td> Robinson </Td> </Tr> <Tr> <Td> 14 </Td> <Td> 607 </Td> <Td> 141053783706...759537328128 </Td> <Td> 366 </Td> <Td> 1952 </Td> <Td> Robinson </Td> </Tr> <Tr> <Td> 15 </Td> <Td> 1,279 </Td> <Td> 541625262843...764984291328 </Td> <Td> 770 </Td> <Td> 1952 </Td> <Td> Robinson </Td> </Tr> <Tr> <Td> 16 </Td> <Td> 2,203 </Td> <Td> 108925835505...834453782528 </Td> <Td> 1,327 </Td> <Td> 1952 </Td> <Td> Robinson </Td> </Tr> <Tr> <Td> 17 </Td> <Td> 2,281 </Td> <Td> 994970543370...675139915776 </Td> <Td> 1,373 </Td> <Td> 1952 </Td> <Td> Robinson </Td> </Tr> <Tr> <Td> 18 </Td> <Td> 3,217 </Td> <Td> 335708321319...332628525056 </Td> <Td> 1,937 </Td> <Td> 1957 </Td> <Td> Riesel </Td> </Tr> <Tr> <Td> 19 </Td> <Td> 4,253 </Td> <Td> 182017490401...437133377536 </Td> <Td> 2,561 </Td> <Td> 1961 </Td> <Td> Hurwitz </Td> </Tr> <Tr> <Td> 20 </Td> <Td> 4,423 </Td> <Td> 407672717110...642912534528 </Td> <Td> 2,663 </Td> <Td> 1961 </Td> <Td> Hurwitz </Td> </Tr> <Tr> <Td> 21 </Td> <Td> 9,689 </Td> <Td> 114347317530...558429577216 </Td> <Td> 5,834 </Td> <Td> 1963 </Td> <Td> Gillies </Td> </Tr> <Tr> <Td> 22 </Td> <Td> 9,941 </Td> <Td> 598885496387...324073496576 </Td> <Td> 5,985 </Td> <Td> 1963 </Td> <Td> Gillies </Td> </Tr> <Tr> <Td> 23 </Td> <Td> 11,213 </Td> <Td> 395961321281...702691086336 </Td> <Td> 6,751 </Td> <Td> 1963 </Td> <Td> Gillies </Td> </Tr> <Tr> <Td> 24 </Td> <Td> 19,937 </Td> <Td> 931144559095...790271942656 </Td> <Td> 12,003 </Td> <Td> 1971 </Td> <Td> Tuckerman </Td> </Tr> <Tr> <Td> 25 </Td> <Td> 21,701 </Td> <Td> 100656497054...255141605376 </Td> <Td> 13,066 </Td> <Td> 1978 </Td> <Td> Noll & Nickel </Td> </Tr> <Tr> <Td> 26 </Td> <Td> 23,209 </Td> <Td> 811537765823...603941666816 </Td> <Td> 13,973 </Td> <Td> 1979 </Td> <Td> Noll </Td> </Tr> <Tr> <Td> 27 </Td> <Td> 44,497 </Td> <Td> 365093519915...353031827456 </Td> <Td> 26,790 </Td> <Td> 1979 </Td> <Td> Nelson & Slowinski </Td> </Tr> <Tr> <Td> 28 </Td> <Td> 86,243 </Td> <Td> 144145836177...957360406528 </Td> <Td> 51,924 </Td> <Td> 1982 </Td> <Td> Slowinski </Td> </Tr> <Tr> <Td> 29 </Td> <Td> 110,503 </Td> <Td> 136204582133...233603862528 </Td> <Td> 66,530 </Td> <Td> </Td> <Td> Colquitt & Welsh </Td> </Tr> <Tr> <Td> 30 </Td> <Td> 132,049 </Td> <Td> 131451295454...491774550016 </Td> <Td> 79,502 </Td> <Td> </Td> <Td> Slowinski </Td> </Tr> <Tr> <Td> 31 </Td> <Td> 216,091 </Td> <Td> 278327459220...416840880128 </Td> <Td> 130,100 </Td> <Td> 1985 </Td> <Td> Slowinski </Td> </Tr> <Tr> <Td> 32 </Td> <Td> 756,839 </Td> <Td> 151616570220...600565731328 </Td> <Td> 455,663 </Td> <Td> </Td> <Td> Slowinski & Gage </Td> </Tr> <Tr> <Td> 33 </Td> <Td> 859,433 </Td> <Td> 838488226750...540416167936 </Td> <Td> 517,430 </Td> <Td> </Td> <Td> Slowinski & Gage </Td> </Tr> <Tr> <Td> 34 </Td> <Td> 1,257,787 </Td> <Td> 849732889343...028118704128 </Td> <Td> 757,263 </Td> <Td> </Td> <Td> Slowinski & Gage </Td> </Tr> <Tr> <Td> 35 </Td> <Td> 1,398,269 </Td> <Td> 331882354881...017723375616 </Td> <Td> 841,842 </Td> <Td> </Td> <Td> Armengaud, Woltman, et al . </Td> </Tr> <Tr> <Td> 36 </Td> <Td> 2,976,221 </Td> <Td> 194276425328...724174462976 </Td> <Td> 1,791,864 </Td> <Td> </Td> <Td> Spence, Woltman, et al . </Td> </Tr> <Tr> <Td> 37 </Td> <Td> 3,021,377 </Td> <Td> 811686848628...573022457856 </Td> <Td> 1,819,050 </Td> <Td> 1998 </Td> <Td> Clarkson, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 38 </Td> <Td> 6,972,593 </Td> <Td> 955176030521...475123572736 </Td> <Td> 4,197,919 </Td> <Td> 1999 </Td> <Td> Hajratwala, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 39 </Td> <Td> 13,466,917 </Td> <Td> 427764159021...460863021056 </Td> <Td> 8,107,892 </Td> <Td> </Td> <Td> Cameron, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 40 </Td> <Td> 20,996,011 </Td> <Td> 793508909365...578206896128 </Td> <Td> 12,640,858 </Td> <Td> 2003 </Td> <Td> Shafer, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 41 </Td> <Td> 24,036,583 </Td> <Td> 448233026179...460572950528 </Td> <Td> 14,471,465 </Td> <Td> </Td> <Td> Findley, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 42 </Td> <Td> 25,964,951 </Td> <Td> 746209841900...874791088128 </Td> <Td> 15,632,458 </Td> <Td> 2005 </Td> <Td> Nowak, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 43 </Td> <Td> 30,402,457 </Td> <Td> 497437765459...536164704256 </Td> <Td> 18,304,103 </Td> <Td> 2005 </Td> <Td> Cooper, Boone, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 44 </Td> <Td> 32,582,657 </Td> <Td> 775946855336...476577120256 </Td> <Td> 19,616,714 </Td> <Td> 2006 </Td> <Td> Cooper, Boone, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 45 </Td> <Td> 37,156,667 </Td> <Td> 204534225534...975074480128 </Td> <Td> 22,370,543 </Td> <Td> 2008 </Td> <Td> Elvenich, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 46 </Td> <Td> 42,643,801 </Td> <Td> 144285057960...837377253376 </Td> <Td> 25,674,127 </Td> <Td> 2009 </Td> <Td> Strindmo, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 47 </Td> <Td> 43,112,609 </Td> <Td> 500767156849...221145378816 </Td> <Td> 25,956,377 </Td> <Td> 2008 </Td> <Td> Smith, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 48 </Td> <Td> 57,885,161 </Td> <Td> 169296395301...626270130176 </Td> <Td> 34,850,340 </Td> <Td> 2013 </Td> <Td> Cooper, Woltman, Kurowski, et al . </Td> </Tr> <Tr> <Td> 49 </Td> <Td> 74,207,281 </Td> <Td> 451129962706...557930315776 </Td> <Td> 44,677,235 </Td> <Td> 2016 </Td> <Td> Cooper, Woltman, Kurowski, Blosser, et al . </Td> </Tr> <Tr> <Td> 50 </Td> <Td> 77,232,917 </Td> <Td> 109200152134...402016301056 </Td> <Td> 46,498,850 </Td> <Td> 2017 </Td> <Td> Pace, Woltman, Kurowski, Blosser, et al . </Td> </Tr> </Table> <Tr> <Th> Rank </Th> <Th> p </Th> <Th> Perfect number </Th> <Th> Digits </Th> <Th> Year </Th> <Th> Discoverer </Th> </Tr>

List of perfect numbers from 1 to 100