Perfect numbers
Perfect numbers are closely related to Mersenne primes. The Greek mathematician Euclid
pointed out that when 2n-1 is a prime, then 2n-1 (2n – 1) becomes a perfect number.
It means that Mersenne prime is a factor to its corresponding perfect number.
Table.11.Mersenne prime and perfect number
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Prime 2p – 1 2p-1(2p – 1) factors
Number p Mersenne prime perfect number
___________________________________________________________ 2 3 6 3 X 2
3 7 28 7 X 4
5 31 496 31 X 16
7 127 8128 127 X 64
13 8191 33550336 8191 X 4096
17 131071 8589869056 131071 X 65536
19 524287 137438691328 524287 X 262144
31 2147483647 2305843008139952128 .........................
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The 24 th Mersenne prime is 2^19937 – 1 and the perfect number associated with this is
2^19936 (2^19937 – 1) This is one of the largest perfect numbers. It has 12,003 digits and begins and ends with 931,144.... 942,656 respectively and found by Bryant Tuckerman in 1971.
2^19936 (2^19937 – 1) This is one of the largest perfect numbers. It has 12,003 digits and begins and ends with 931,144.... 942,656 respectively and found by Bryant Tuckerman in 1971.
Perfect number is a number which can be expressed as the sum of its all divisors except that
number itself. For example, 6,28,496,8128,33550336 .... are first few perfect numbers.
It can be shown that they are all equal to sum of their divisors.
6 = 1 + 2 + 3
28 = 1 + 2 +4 + 7 +14
496 = 1 + 2 + 4 + 8 +16 + 31 +62+ 124 + 248
8128 = 1+2+4+8+16+32+64+127+508+1016+2032+4064
33550336 = 1+2+4+8+16+32+64+128+256+512+1024+2048+4096+
8191+16382+32764+131056+262112+524224+
1048448+20966896+4193792+8387584+16775168
The first 30 perfect numbers are given in Table.12.
Table.12. First 30 perfect numbers
__________________________________________________________________________No Perfect number number of digits
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1 21(22 – 1) = 6 1
2 22(23 – 1) = 28 2 3 24(25 - 1) = 496 3
4 26(27 – 1) = 8128 4
5 212(213 - 1) = 33,550,336 8
6 216(217 – 1) = 8,589,869,056 10
7 218(219 - 1) = 137,438,691,328 12
8 230(231 – 1) = 2,305,843,008,139,952,128 19
9 260(261 – 1) 37
10 288(289 – 1) 54
11 2106(2107 – 1) 65
12 2126( 2127 – 1) 77
13 2520 (2521 – 1) 314
14 . 2606(2607 – 1 ) 366
15. 21278(21279 – 1) 770
16. 22202 (22203 – 1) 1327
17 . 22280(22281 – l) 1373
18. 23216(23217 – 1) 1937
19. 24252(24253 – 1) 2561
20. 24422(24423 – 1) 2663
21. 29688(29689 – 1) 5834
22. 29940(29941 – 1) 5985
23. 211212(211213 – 1) 6751
24. 219936(219937 – 1) 12003
25. 221700(221701 – 1) 13006
26. 223208(223209 – 1) 13973
27. 244496(244497 – 1) 26790
28. 286242 (286243- 1) 51924
29. 2132048(2132049 – 1) 79502
30. 2216090(2216091 – 1) 130100
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Mersenne prime and perfect number from chess board
Mersenne primes and perfect numbers can be easily generated on our chess board. Once upon a Time, a chess competition between an Indian king and a farmer was conducted. The condition was, if the king wins, the farmer will be beheaded, and if the king is defeated, he has to pay some amount of grains, which the farmer wanted. The quantity of grains must be estimated according to 1 grain in the first square, 2 in the second, 4 in the third, 16 in the fourth, 256 in the fifth and so on. At the end of the game, the king was defeated. When he tried to pay the grains, he actually felt that even the total grains preserved in his whole kingdom could not fulfil the farmer’s condition.
To obtain a Mersenne prime from the chess board, remove one grain from the grains on a square and check if the number of the remaining gains is a prime. If it is, then it is a Mersenne prime .Also the product of the number of grains in this square and that of those in the previous square will give a perfect number.
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Some properties of perfect numbers
The perfect numbers have some common properties. They are
1. All known perfect numbers are even. No odd perfect numbers have been
identified. Thus all perfect numbers are divisible by2. A number to be even,
it should end with 2,4,6,8 or 0. However, all perfect numbers end only with
either 6 or 8. They do not end with 0 or 2 or 4.
identified. Thus all perfect numbers are divisible by
it should end with 2,4,6,8 or 0. However, all perfect numbers end only with
either 6 or 8. They do not end with 0 or 2 or 4.
2. The factors of a perfect number will not be in natural series. But a perfect
number can be expressed as a sum of all the successive numbers from 1 up
to a prime factor of the perfect number. e.g.,
number can be expressed as a sum of all the successive numbers from 1 up
to a prime factor of the perfect number. e.g.,
6 = 1 + 2 + 3 = (3x4)/2
28 = 1 + 2+ 3 + 4 + 5+6 + 7 = (7x8)/2
496 = 1 + 2 +3 + 4 + 5+....... + 31 = (31x32)/2
8128 = 1 + 2 +3 + 4 + 5+ ..... + 127 = (127 x128)/2
3. The square of a number which is equal to a sum of all number up to a number
in the natural series, can be expressed as the sum of cube of numbers in natural series .
e.g.,
in the natural series, can be expressed as the sum of cube of numbers in natural series .
e.g.,
62 = 13 + 23 + 33
282 = 13 + 23 + 33 + 43 + 53 + 63 + 73
4962 = 13 + 23 + 33 + 43 + 53 + ..... + 313
81282 = 13 + 23 + 33 + 43 + 53 + ..... + 1273
4. All perfect numbers except 6 can be represented as a sum of cube of
successive odd numbers in the natural series from 1 ,e.g.,
successive odd numbers in the natural series from 1 ,e.g.,
28 = 13 + 33 = 4 x 7 = 22 x 7 ; (1 + 3 = 4)
496 = 13 + 33 + 53 + 73 = 16 x 31 = 24 x 31 ; (1+3+5+7= 16)
8128 = 13 + 33 + 53 + 73 + 93 + 113 + 133 + 153 = 64 x 127
= 26 x 127 ;
= 26 x 127 ;
(1+3+5+7+9+11+13+15 = 64)
5. Perfect numbers can also be expressed as a sum of successive powers of 2 e.g.,
6 = 21 + 22
28 = 22 + 23 + 24
496 = 24 + 25 + 26 + 27 + 2 8
8128 = 26 + 27 + 28 + 29 + 210 + 211 + 212
6. Except 6, all other perfect numbers have its digital root l. The digital root of a
number is a single digit number obtained by summing up all of the digits of the number. e.g.,
number is a single digit number obtained by summing up all of the digits of the number. e.g.,
28 ; 2+8 = 10 ; 1
496; 4 + 9 +6 = 19 ; 1+9 = 10 ; 1
8128; 8+1+2+8 = 19 ; 1+9 = 10 ; 1
7. Reciprocals of the divisors of a perfect number including the number itself always add up to 2 e.g.,
6 ; (l/1) + (1/2) + (1/3) + (1/6) = 2
28 ; (1/1) +(1/2)+(1/4) + (1/7) +(1/14)+(1/28) = 2
8.A perfect symmetry is observed among the divisors of the perfect numbers. Few
simple perfect numbers and their divisors are listed below.
simple perfect numbers and their divisors are listed below.
6 : 1,2,(22 – 1)
28 : 1,2,22, (23 – 1), 2(23 – 1)
496 : 1,2,22,23,24, (25 – 1),2(25 - 1), 22 (25 – 1),23(25 – 1)
8128 : 1,2,22,23,24,25,26, (27 –l),2(27 – 1),22(27 –l),23(27 –l),24(27 -1),25(27 – 1)
From this we understand that the first four perfect numbers 6,28,496 and 8128
are related To the prime numbers 2,3,5 and 7 respectively. According to this
correlation, if p is a prime number, then its related perfect number will have
factors 1,2,22,23,....... 2p-1,(2p – 1),2(2p – 1),22(2p -1)..................2p-2(2p – 1).
are related To the prime numbers 2,3,5 and 7 respectively. According to this
correlation, if p is a prime number, then its related perfect number will have
factors 1,2,22,23,....... 2p-1,(2p – 1),2(2p – 1),22(2p -1)..................2p-2(2p – 1).
9. All perfect numbers end with either 6 or 8 and no perfect number is odd.
This can be established mathematically. If the perfect number is odd, it will
not have any single even factor, because the product of such factor with other
factors, whatever they may be will result with even number. Hence odd
perfect number, if exists will not have even factors at all. The factors of a number
can be paired up. When the number is divided by one of its factors in a pair, the
other factor of the pair will result. If 1 is a factor, then its pairing factor will be
the number itself .Let us suppose that an odd perfect number has a pair of factors
a and b. Then
This can be established mathematically. If the perfect number is odd, it will
not have any single even factor, because the product of such factor with other
factors, whatever they may be will result with even number. Hence odd
perfect number, if exists will not have even factors at all. The factors of a number
can be paired up. When the number is divided by one of its factors in a pair, the
other factor of the pair will result. If 1 is a factor, then its pairing factor will be
the number itself .Let us suppose that an odd perfect number has a pair of factors
a and b. Then
P = a x b = 1 x P
Where P denotes odd perfect number and a and b are odd factors. But as per
definition P must be equal to the sum of all of its factors except P
definition P must be equal to the sum of all of its factors except P
P = 1 + a + b = 1 + a + P/a
Which gives P = a(a+1)/(a-1) and b = (a+1)/(a-1). For all possible odd values
of a, b cannot be odd and integer. It implies that odd perfect number is not
possible. 10.An even perfect number p can be related directly to its corresponding
Mersenne prime M = 2p - 1.
of a, b cannot be odd and integer. It implies that odd perfect number is not
possible. 10.An even perfect number p can be related directly to its corresponding
Mersenne prime M = 2p - 1.
p = (½)(M)(M+1)
11.The number digit of a prime number can be a perfect number. The least prime
number whose number digit is a perfect number 28 is 1999.
number whose number digit is a perfect number 28 is 1999.
1+9+9+9 = 28 = 1+2+4+7+14
It is noted that there is no prime numbers whose sum of digits is equal to 6.Because
it will be divisible by 3 and hence it cannot be prime in nature. The least prime whose
number digit is the next perfect number 496 is
it will be divisible by 3 and hence it cannot be prime in nature. The least prime whose
number digit is the next perfect number 496 is
2 999 999 999 999 999 999 999 9 8 999 999 999 999 999 999 999 999 999 999 99
2(22 nines) 8 (32 nines)
It is a challenging entertainment to find primes whose number digit is a perfect number.
In fact there are quite a lot of such primes. If we restrict ourselves with multi-digit
primes using only two digits, the challenge becomes more thrilling. Few examples for
the perfect number 28 are
In fact there are quite a lot of such primes. If we restrict ourselves with multi-digit
primes using only two digits, the challenge becomes more thrilling. Few examples for
the perfect number 28 are
1181881 2222929 338383
1881181 2922229 383833
1881811
8118181
8188111
