A perfect number is the sum of its proper divisors , by definition . Perfect Numbers = Perfect Integers = Perfect Naturals = Perfects p p-1 p 2 - 1 is prime ===> P = 2 ( 2 - 1 ) is perfect p E.g. , p = 3 , 2 - 1 = 7 , P = 1 + 2 + 4 + 7 + 14 = 28 . p Every even perfect is of this form , where P = Triangle ( 2 - 1 ) ; e.g. , 28 = 7 + 6 + 5 + 4 + 3 + 2 + 1 . They may be infinite in number ; 48 are known as of 2013-02-05 . p 2 - 1 is prime ===> p is prime No odd perfect is known . p In binary radix representation , 2 - 1 is a repunit , e.g. , 11 , 111 , 11111 , 1111111 . In radix-2 , every even perfect is a string of p-1 0s appended to a string of p 1s : e.g. , 110 , 11100 , 111110000 , 1111111000000 . The sum of the reciprocals of the divisors > 1 of an even perfect is 1 ; e.g. , 1/2 + 1/3 + 1/6 = 1 , 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1 . P mod 10 = 6 or 8 Let P be even perfect > 6 : P mod 9 = 1 P mod 12 = 4 P mod 6 = 4 P mod 3 = 1 (p-1)/2 2 ____ p-1 p \ 3 P = 2 ( 2 - 1 ) = > ( 2i - 1 ) /___ 1 3 3 3 3 e.g. , 496 = 1 + 3 + 5 + 7 Elementary functions of even perfects sum of divisors (P) Euler's phi (P) # divisors (P) abundancy (P) p p p-2 p 2p 2 ( 2 - 1 ) 2 ( 2 - 2 ) 2 --- Perfect Naturals are not Perfect Pow-ers ( e.g. , perfect squa-res , perfect cu-bes , etc. ) --- Odd Perfect Natural = OPN (??) q[1] q[2] q[r] p[1] * p[2] * ... * p[r] = OPN , where the p[i] are distinct primes p[r] >= p[i] p[1] mod 4 = 1 q[1] mod 4 = 1 For i > 1 , q[i] mod 2 = 0 OPN mod 36 = 1 or 9 or 13 or 25 300 OPN > 10 r 4 2 > OPN r > 8 p[r] > 100000000 q[i] 20 p[i] > 10 , for some i p[r-1] > 10000 p[r-2] > 100 ____ \ > q[i] > 74 /___ r >= p[1] --- Mersenne p p-1 p rank p prime 2 - 1 Perfect 2 ( 2 - 1 ) 1 2 3 6 2 3 7 28 3 5 31 496 4 7 127 8128 5 13 8191 33550336 6 17 131071 8589869056 7 19 524287 137438691328 8 31 2147483647 2305843008139952128 9 61 2305843009213693951 2658455991569831744654692615953842176 10 89 11 107 . . 12 127 . . 13 521 . . 14 607 15 1279 16 2203 17 2281 18 3217 19 4253 20 4423 21 9689 22 9941 23 11213 24 19937 25 21701 26 23209 27 44497 28 86243 29 110503 30 132049 31 216091 32 756839 33 859433 34 1257787 35 1398269 36 2976221 37 3021377 38 6972593 39 13466917 40 20996011 41 24036583 42 25964951 ? 30402457 ? 32582657 ? 37156667 ? 42643801 ? 43112609 ?? 57885161 By orders of magnitude = # of digits : 1 2 3 4 5 6 7 8 4 6 4 8 6 5 5 10+ --- metas : elementary number theory multiplicative number theory additive number theory sigma abundancy integer natural number sum of proper divisors Mersenne prime perfect number perfect numbers perfect integer perfect integers perfects 2-perfects nombre parfait nombres parfaits entier parfait entiers parfaits vollkommene Zahl vollkommene Zahlen vollkommenen Zahlen vollkommener Zahlen perfekte Zahl perfekte Zahlen vollkommene Ganzzahl vollkommene Ganzzahlen perfekte Ganzzahl perfekte Ganzzahlen perfekt tal perfekte tal fuldkomne tal n'umero perfecto numero perfecto n'umeros perfectos numeros perfectos entero perfecto enteros perfectos numerus perfectus numeri perfecti aliquot divisors vollkommen vollst"andig vollstandig deficient diminute defective unvollkommen unvollst"andig unvollstandig mangelhaft abundant superfluos plus quam-perfectus redundantem exc'edant excedant "ubervollst"andig ubervollstandig "uberflussig uberflussig "uberschiessende uberschiessende tochnyi chislo (??) tochnyi chisel (??) --- Mathematicians and science writers , please take note : + + For the integers , Z = J , the terms "divisor" and "factor" are not interchangable as they are in ring theory ( more precisely , ideal theory ) . Factors form a product . Free-standing integers are merely divisors . These terms are well-formed : divisor , factor , divisors , factors , the greatest common divisor , the set of divisors , the set of positive divisors , the set of proper divisors , sum of the proper divisors , the set of prime factors , greatest common prime factor , the unique set of prime factors , a set of factors , the sum of all its proper divisors . These terms are ill-formed : the set of factors , sum of the factors , greatest common factor . These terms are incorrect : the sum of all the divisors , the sum of all its factors . E.g. , if the factors of 28 are purportedly 1 , 2 , 4 , 7 , and 14 , then 1 * 2 * 4 * 7 * 14 = 784 , thus exhibiting 1 , 2 , 4 , 7 , and 14 as a set of factors of 784 . ---It takes no special mathematical nor computing skills to find the next perfect number .
2010 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11A63 , 11N05 , 11N25 , 11Y05 , 11Y11 , 11Y55 , 11Y70
Walter Nissen
posted 2008-03-05 updated 2013-02-05