Perfect number

Illustration through the Cuisenaire regettes that the 6 has the properties of a perfect number

A perfect number is a positive integer that is equal to the sum of its positive proper divisors. In other words, a perfect number is one that is a friend of itself.

Thus, 6 is a perfect number because its positive proper divisors are 1, 2 and 3; and 6 = 1 + 2 + 3. A positive proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper factors of 6 are 1, 2, and 3, but not 6. The next perfect numbers are 28, 496, and 8128.

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 + 254 + 508 + 1016 + 2032 + 4064

History

The mathematician Euclides discovered that the first four perfect numbers are given by the formula 2n− − 1⋅ ⋅ (2n− − 1){displaystyle 2^{n-1}cdot (2^{n}-1)}:

n = 2:2¹ × 2² - 1) = 6

n = 3:2² × 2³ - (1) = 28

n = 5: 2⁴ × 2⁵ - 1) = 496

n = 7: 2⁶ × 2⁷ - (1) = 8128

Realizing that 2ⁿ – 1 is a prime number in every case, Euclid proved that the formula 2^n–1(2ⁿ – 1) generates an even perfect number whenever 2ⁿ – 1 is prime.

Ancient mathematicians made many assumptions about perfect numbers based on the four they already knew. Many of these assumptions have turned out to be false. One was that since 2, 3, 5, and 7 were precisely the first four prime numbers, the fifth perfect number would be obtained with n = 11, the fifth prime number. However, 2¹¹ – 1 = 2047 = 23 × 89 is not prime and therefore n = 11 does not generate a perfect number. Two of the other wrong assumptions were:

1. The perfect fifth number would have five digits, since the first four have 1, 2, 3 and 4, respectively.
2. The perfect numbers would end alternatively in 6 and 8.

The fifth perfect number (33 550 336) has 8 digits, thus contradicting the first assumption. As for the second, the fifth perfect number ends in 6, but also the sixth (8 589 869 056) ends in 6. (The fact that the last digit of an even perfect number expressed in base 10 is always 6 or 8 is not hard to prove).

It was in 1603 when Pietro Cataldi found the perfect numbers sixth and seventh, 2¹⁶(2¹⁷ – 1) = 8 589 869 056 and 2¹⁸ (2¹⁹ – 1)= 137 438 691 328.

It is true that if 2ⁿ – 1 is a prime number, then n must also be prime, but the reciprocal is not necessarily TRUE. Today, the prime numbers generated by the formula 2ⁿ – 1 are known as Mersenne primes, after the 17th-century monk Marin Mersenne, who studied number theory and perfect numbers.

Subsequently, Leonhard Euler demonstrated in the 18th century that all even perfect numbers are generated from the formula already discovered by Euclid: the Euclid-Euler Theorem.

Odd perfect numbers are not known to exist. However, there are some partial results in this regard. If there exists an odd perfect number it must be greater than 10³⁰⁰, it must have at least 8 distinct prime factors (and at least 11 if it is not divisible by 3). One of those factors must be greater than 10⁷, two of them must be greater than 10,000, and three factors must be greater than 100.

On December 7, 2018, upon discovery of the largest prime number 2^{82 589 933} − 1 (or M_{82 589 933} in the usual notation), the largest perfect number found to date was then obtained, number 51 on the list, with 49,724,095 digits:

The aforementioned cousin was discovered by Patrick Laroche as part of the Great Internet Mersenne Prime Search (GIMPS) project.

Other properties of even perfect numbers

They are triangular numbers

A triangular number is shaped n2+n2{displaystyle {frac {n^{2} +n}{2}}}}}where «n» is any positive integer different from zero. If we leave the identity 2p− − 1(2p− − 1)=(2p− − 1)+12(2p− − 1){displaystyle 2^{p-1}left(2^{p}-1right)={frac {left(2^{p}-1right)+1}{2}{2}}left(2^{p}-1right)} and distribute the product of the second member we get:

2p− − 1(2p− − 1)=(2p− − 1)2+(2p− − 1)2{displaystyle 2^{p-1}left(2^{p}-1right)={frac {left(2^{p}-1right)^{2}+left(2^{p}-1right)}{2}}{2}}}}{2}}.

The expression 2p− − 1{displaystyle 2^{p}-1} it is a prime number of Mersenne and we see that the right term of identity adopts the form corresponding to the definition of triangular number. We can say that a perfect number pair is a triangular number and its order is a prime number of Mersenne.

They are combinatorial numbers or coefficients of the binomial

As all triangular numbers are in the third column of the triangle of Pascal and we have just seen that all perfect number pair is a triangular number, the perfect numbers are also combinatorial numbers. (2p2){displaystyle textstyle {2^{p} choose 2}}}Where 2p{displaystyle 2^{p}} is the power corresponding to a prime number of Mersenne increased in one unit.

These are hexagonal numbers

A hexagonal number is shaped n(2n− − 1)=2n2− − n{displaystyle n(2n-1)=2n^{2}-n}for “n» any positive integer different from zero. It emerges immediately from the identity 2p− − 1(2p− − 1)=2p− − 1(2⋅ ⋅ 2p− − 1− − 1){displaystyle 2^{p-1}left(2^{p}-1right)=2^{p-1}left(2cdot 2^{p-1}-1right)}}calling "n» to number 2p− − 1{displaystyle 2^{p-1}.

Open questions

An open question is understood as a property for which there is still no proof, both of its affirmation and its denial. They are open questions:

• Determine if there are infinite perfect numbers. Until December 2018 51 perfect numbers are known.
• Demonstrate the impossibility of a perfect number odd or find one.
• The calculation of the number of dividers and the sum of their dividers can be found by a characteristic issue of power of a number, for example if a positive real number rises to the square always increases its number, on the other hand the numbers from 0 to 1 without counting the numbers always tend to decrease being the zero limit and for the numbers 1 tend to 1 and finally for the numbers from 1 onward tend to the infinite. Thanks to this feature, there is a function of making such calculations of dividers, sum and quantity. The function is equal to 2 for the prime numbers, the function for the primality study using that characteristic.

Related concepts

Besides, and considering the sum of the positive divisors, there are other types of numbers.

• Defective numbers: the sum of positive dividers is lower than the number.
• Abundant numbers: the sum is greater than the number.
• Number of friends: a and b such that a is the sum of the positive dividers b and vice versa.
• Sociable numbers: like friends, but with a larger cycle of numbers.
• Semi-perfect numbers: the sum of all or some of the positive dividers is equal to the number.