Mystery of ‘Perfect Numbers’ Resolved – Perfect Number is Always Even and Predictable

Vrajlal Sapovadia (Ph.D.)
Sweta Patel (Ph.D.)

Introduction

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. In other words, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n. To explain in practical terms, we elaborate first few Perfect Numbers. It may be noted that ‘Perfect Numbers’ are sparse are thinly dispersed. Starting from 3^rd Century BC, mathematicians are working on Perfect Numbers. Till April 2018, i.e. during last 2300 years active research, researchers could recognize only 50 perfect numbers. There are 2 perfect numbers in first 100 and 4 in first million. Absolute distance between two perfect numbers increase exponentially as you go higher to the next perfect number[1]. One can find at least one perfect number till 4 digit numbers, and then it becomes a real rarity. Subsequent perfect numbers appears at 8, 10, 12 and 19 digits. 15^th perfect number has 770 digits while 16^th have 1327 digits. 25^th perfect number has 13066 digits. 50^th perfect number has 46,498,850 digits.

The current literature is still debating on two issues:

1. Can perfect number is predictable?
2. Can perfect number be odd?

We argue that perfect number is predictable and we have developed a formula which answers both lead questions as follow:

1. Perfect number is predictable
2. Perfect number is always even

Predictability

Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Euclid, Prop. IX.36). the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number, as follows:

for p = 2:   21(22 − 1) = 6

for p = 3:   22(23 − 1) = 28

for p = 5:   24(25 − 1) = 496

for p = 7:   26(27 − 1) = 8128.

Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number.[11] In fact, Mersenne primes are very rare—of the 2,270,720 prime numbers p up to 37,156,667,[12] 2p − 1 is prime for only 45 of them.

Nicomachus (60–120 AD) conjectured that every perfect number is of the form 2p−1(2p − 1) where 2p − 1 is prime.[13] Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of that form.[14] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem. As of January 2018, 50 Mersenne primes are known,[15] and therefore 50 even perfect numbers (the largest of which is 277232916 × (277232917 − 1) with 46,498,850 digits).

Owing to their form, 2p−1(2p − 1), every even perfect number is represented in binary as p ones followed by p − 1  zeros. Interestingly, when a perfect number is converted into binary, it is not only a pernicious number, but binary sequence is spectacular having all 1 on the left side followed by all 0. Interestingly count of 1 is a prime number (p) and 0 is p-1.

6₁₀ = 110₂

1 (p = 2) and 0 (1)

28₁₀ = 11100₂

1 (p = 3) 0 (2)

496₁₀ = 111110000₂

1 (p = 5) 0 (4)

8128₁₀ = 1111111000000₂

1 (p = 7) 0 (6)

33550336₁₀ = 1111111111111000000000000₂

1 (p = 13) 0 (12)

8589869056₁₀ = 111111111111111110000000000000000₂

1 (p = 17) 0 (16)

137438691328₁₀ = 1111111111111111111000000000000000000₂

1 (p = 19) 0 (18)

2305843008139952128₁₀ = 1111111111111111111111111111111000000000000000000000000000000₂

1 (p = 31) 0 (30)

Thus every even perfect number is a pernicious number. Note that every even perfect number is also a practical number. Therefore a formula to find a perfect number can be developed as 1….(p) 0…(p-1), where 1 (p) and 0 (p-1) are binary symbol. Thus, a binary number so written equal to a PRIME (p) ‘1’ followed by p-1 ‘0’ could be a perfect number. It may be noted that all prime count does not result into perfect number. Therefore, it is pertinent to test each prime number with formula[2] will establish whether resultant number is perfect number or not. But in any case, this will reduce substantially the experiment time to find next perfect number or this formula provides a lead to find perfect number with less experiment time.

Odd vs. Even

It is unknown whether there is any odd perfect number, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid’s rule gives all perfect numbers, thus implying that no odd perfect number exists. More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also Ore’s harmonic numbers, and it has been conjectured as well that there are no odd Ore’s harmonic numbers other than 1. An exhaustive search by the GIMPS[3] distributed computing project has shown that the first 46 are all even numbers represented by 2p−1(2p − 1).

First 50 perfect numbers listed[4] are all even and their last one or two digits are always 6 or 28. In support of arguments made by various researchers, we found that a perfect number can be presented as binary with formula 1….(p) 0…(p-1). Any binary pattern as 1 (n) 0 (n-1) will always result into even number. Therefore any perfect number is always a even number.

[1] 6 (1), 28 (2), 496 (3), 8128 (4), 33550336 (8), 8589869056 (10), 137438691328 (12), 2305843008139952128 (19)
[2] Binary numbers 1….(p) 0…(p-1)
[3] The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
[4] https://en.wikipedia.org/wiki/List_of_perfect_numbers