Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹ (2ⁿ − 1):

For n = 2: 21(22 − 1) = 6

For n = 3: 22(23 − 1) = 28

For n = 5: 24(25 − 1) = 496

For n = 7: 26(27 − 1) = 8128

Noticing that 2n − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹ (2ⁿ − 1) gives an even perfect number whenever 2ⁿ − 1 is prime. Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2¹¹ − 1 = 2047 = 23 • 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:

• The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.

• The perfect numbers would alternately end in 6 or 8.

The fifth perfect number (33550336 = 2¹²(2¹³ − 1)) has 8 digits, thus debunking the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show the last digit of any even perfect number must be 6 or 8.

In order for 2ⁿ − 1 to be prime, it is necessary that n should be prime. Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.

Two millennia after Euclid, Euler proved that the formula 2ⁿ⁻¹ (2ⁿ − 1) will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the “Euclid-Euler Theorem”. As of October 2006 only 44 Mersenne primes are known, which means there are 44 perfect numbers known, the largest being 2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits. It is still uncertain whether there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS (Great Internet Mersenne Prime Search) distributed computing project.

Since any even perfect number has the form 2ⁿ⁻¹(2ⁿ − 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2ⁿ − 1. Furthermore, any even perfect number except the first one is the sum of the first 2^(n−1)/2 odd cubes:

$6 = 2^1(2^2-1) = 1+2+3, \,$

$28 = 2^2(2^3-1) = 1+2+3+4+5+6+7 = 1^3+3^3, \,$

$496 = 2^4(2^5-1) = 1+2+3+\cdots+29+30+31 = 1^3+3^3+5^3+7^3, \,$

Next let us talk on Odd Perfect numbers, Triangular numbers.

Note: A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n.

Do you want a table of prime factors of 1 to 1000.