8128
nobody is perfect - numbers can be, namely: 6, 28, 496, 8128, .....
definition of a perfect number: sum of its divisors gives the number itself
for example divisors of 8128:
1,2,4,8,16,32,64,127,254,508,1016,2032,4064
1+2+4+8+16+32+64+127+254+508+1016+2032+4064 = 8128
it is supposed that all perfect numbers are even
then, for n > 6, all perfect numbers are
the sum of odd cubes:
28 = 1^3 + 3^3
496 = 1^3 + 3^3 + 5^3 + 7^3
8128 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3
.....
and they can be written as
n = 1 + 9*k*(k+1)/2 , k = 8j+2, j not negative integer
28 = 1 + 9*2*(2+1)/2 (j=0)
496 = 1 + 9*10*11/2 (j=1)
8128 = 1 + 9*42*43/2 (j=5)
.....
and all as sum of first natural numbers
6 = 1+2+3
28 = 1+2+3+4+5+6+7
496 = 1+2+3+4+5+6+7+8+ ... +31
8128 = 1+2+3+4+5+6+7+8+ ... +127
.....
which is the same as
6 = 3*4/2
28 = 7*8/2
496 = 31*32/2
8128 = 127*128/2
.....
already Euclid knew this formula:
6 = 2^1*(2^2-1)
28 = 2^2*(2^3-1)
496 = 2^4*(2^5-1)
8128 = 2^6*(2^7-1)
in general (proved in 1644 by Marin Mersenne):
perfect number = 2^(n-1) * (2^n -1), where (2^n -1) is prime,
reciprocals of all divisors of a perfect number n (including n) added gives 2
for 8128: 1/1+1/2+1/4+1/8+1/16+1/32+1/64+1/127+1/254+1/508+1/1016+1/2032+1/4064+1/8128 = 2
all perfect numbers end with 6 or 8
their prime factorization has the form 2^i x p, p = 2^(i+1) - 1
the binary code starts with k ones, followed by (k-1) zeros
first 7 perfect numbers (binary):
110
11100
111110000
1111111000000
1111111111111000000000000
111111111111111110000000000000000
1111111111111111111111111111111000000000000000000000000000000
first 11 perfect numbers (decimal):
6
28
496
8 128
33 550 336
8 589 869 056
137 438 691 328
2 305 843 008 139 952 128
2 658 455 991 569 831 744 654 692 615 953 842 176
191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216
13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318 386 943 117 783 728 128
furthermore:
8128 Nicomachus is the name of asteroid 1967JP, in honor of Nicomachus of Gerasa,
Jordan, who lived between 60 and 120 A.D. and who has been credited with the discovery of
8128 to be the fourth perfect number. He was an arithmetician, philosopher and music theoretician.
The asteroid (15 km diameter) was discovered 1967 by Argentinian astronomer Carlos Ulrrico Cesco, the name
8128 Nicomachus was suggested by astronomer B.G. Marsden and is official since 1998.
more interesting 4-digit numbers:
more about artful numbers and their beautiful arithmetic: artmetic
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