NOTE ON PERFECT NUMBERS AND THEIR EXISTENCE

: This paper will address the interesting results on perfect numbers. As we know that, perfect number ends with 6 or 8 and perfect numbers had some special relation with primes. Here one can understand that the reasons of relation with primes and existence of odd perfect numbers. If exists, the structures of odd perfect numbers in modulo .


INTRODUCTION
A perfect number is a whole number in which the term itself is equal to the sum of all its factors. Given below are some example based on perfect numbers. Lets consider the number 28 Factors of 28 are 1, 2, 4, 7, 14 and 28 of this the proper factors are 1, 2, 4, 7, 14 and 28 Some of the proper factors = 1+2+4+7+14 = 28 Hence, 28 is a perfect number. 28 is also considered as the only even perfect number. A perfect number can be defined as an integer, which is a non-zero number. A perfect number can be obtained by adding all the factors which are less than that number. All perfect numbers are even. There is no odd perfect number. It is easy to find unusual properties of small numbers that would characterize these into their peculiarity. The number six (6) has a unique property in which it has both the sum and the product of all its smaller factors; 6 = 1+2+3 or 1 x 2 x 3. By the divisors of a number we mean the factors including unity are less than the number. If the sum of its proper divisors or aliquot divisors is less than the number then we call them as deficient ( as in case of 8). If the total sum of the proper divisors or aliquot divisors exceeds the number as in case of 12, the number is then called abundant. The early Hebrews considered 6 to be a perfect number and Philo Judeus ( 1st century AD) also regarded 6 to be a perfect number. There are two main types of perfect numbers and they are even perfect numbers [4] which would follow 2 n-1 (2 n -1)and odd perfect numbers [2].

EVEN Perfect numbers
Euclid knew that 2 n-1 (2 n -1) was perfect if 2n-1 is prime. Euclid proved that, if and when 'p' = 1 + 2 + 22+ ….2n is a prime then 2 n p is referred as a perfect number. 2 n p is divisible by 1, 2, …2 n , p, 2p ..2 n-1 p is the number less than itself and so the sum of these divisors is 2n p.

ODD Perfect numbers
Eular showed that these numbers have dimensions of p a m 2 , where 'p' is prime and p = a = 1 or mod 4. If 'n' is an odd number with s (n) = an, then n < (4d) 4K , where 'd' is the denominator of a and 'K' is the number of distinct prime factors on 'n'. If n K is an odd number with k distinct prime factors then n < 4 4K Let us discuss our main problems in the following section.

PROBLEMS ON PERFECTNESS
As we want it to be perfect number, we have 2. Every even perfect numbers ends with 6 or 28.
Hence, even perfect numbers ends in 6 or 8. If it ends in 8, we want to show that it is in fact ends in 28. We have that if Now we have; Therefore, every even perfect number ends in 6 or 8. Also, if it ends in 8, then in fact ends in 28. ◊