First let us define a perfect number. We call a number perfect if it is a positive integer and is the sum of all it proper divisors. The proper divisors part means we do not include the number itself even though any number divides itself. So for example 6 is perfect because it is a positive integer and 1+2+3=6. Not we excluded the number 6 even though 6 does divide 6.
Now we could look at any number and find all the divisors of the number except the number itself. Next add them all and if the sum is equal to the number itself. If so then that number is a perfect number. While this method will always work, it is quite tedious.
Euclid found another way thousands of years ago. He discovered that you could find the first four perfect numbers by using the formula 2p-1(2p − 1) where p is a prime such as 2,3,5 or 7. So let us consider the first prime 2 and plug it in Euclid's formula. 22-1 (22 -1)
is equal to 21 (4-1)=2x3=6 which is the first perfect number. It is important to notice that the (2p − 1) portion is always giving us a Prime number. The perfect number we would obtain with the primes 2,3,5, and 7 are also even. For example 23 -1=7 which is prime and we multiply that by 22 so 4x7=28 which is perfect and even.
Everything was great until we tried the 5th prime which is the number 11. When we plug that into the (2p − 1) part just as we did 2 and 3, we get the number 211 − 1 = 2047 = 23 × 89 so it is not a prime number like the values we get with 2,3,5, and 7. Why do we care about the fact the this is not prime? We were looking for perfect numbers not primes?
Because when we multiply 2047 by 211-1 we find out the the number we get is not perfect.
So Euler was correct only for those first few values.
When (2p − 1) is prime it has a special name. It is called a Mersenne prime named after Marin Mersenne who was a monk in the 17th ( a prime number by the way) century who studied math and primes and a field known as number theory. In order for (2p − 1) to be prime is has been found that it is necessary but not sufficient for p to be prime. In math a necessary condition is one that must hold true for the statement to be true. So p must be a prime. However a sufficient condition is one that says if this is satisfied the statement is always true. So the prime 11 showed us the (2p − 1) is not always prime and this means the condition is not sufficient. However, if p is not a prime (2p − 1) will not be prime either.
A century later, Euler proved that all perfect numbers are generated by (2p-1 )(2p − 1).
Euclid has tried but could not prove it. We can find the first 39 perfect numbers by using the primes: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917
Note that all these number are even. We do not know if there are any odd perfect numbers. It has been shown that there are no odd perfect numbers in the interval from 1 to
1050 but we can't say in general.
We don't even know how many perfect numbers or Mersenne primes there are. While we suspect the number is infinite, that has not been proven yet.
So here are the first few perfect numbers and you will see they become quite larger very fast!
6,
28,
496,
8128,
33550336,
8589869056,
137438691328,
2305843008139952128,
2658455991569831744654692615953842176,
191561942608236107294793378084303638130997321548169216,
131640364585696483372397534604587229102234723183869431
17783728128,
144740111546645244279463731260859884815736774914748358
89066354349131199152128,
2356272345726734706578954899670990498847754785839260071014302
7597506337283178622239730365539602600561360255566462503270175
0528925780432155433824984287771524270103944969186640286445341
2803383143979023683862403317143592235664321970310172071316352
7487298747400647801939587165936401087419375649057918549492160
555646976,
1410537837067120690632079580860631898814867435147156678388386
7599995486774265238011410419332903769025156195056870982932716
4087724366370087116731268159313652487450652439805877296207297
4467232951666582288469268077866528701889208678794514783645693
1392206037069506473607357237869517647305526682625328488638371
5072974324463835300053138429460296575143368065570759537328128,
5416252628436584741265446537439131614085649053903169578460392
0818387206994158534859198999921056719921919057390080263646159
2800138276054397462627889030573034455058270283951394752077690
4492443149486172943511312628083790493046274068171796046586734
8720992572190569465545299629919823431031092624244463547789635
4414813917198164416055867880921478866773213987566616247145517
2696430221755428178425481731961195165985555357393778892340514
6222324506715979193757372820860878214322052227584537552897476
2561793951766244263144803134469350852036575847982475360211728
8040378304860287362125931378999490033667394150374722496698402
8240806042108690077670395259231894666273615212775603535764707
9522501738583051710286030212348966478513639499289049732921451
07505979911456221519899345764984291328
You can find every know Mersenne prime and hence every know perfect number on a site called GIMPS. I have included a link. But even more than seeing them, you can help discover the next one and become quite famous!
Now this nice answer on perfect numbers will conclude with an important theorem.
If 2k-1 is a prime number, then 2k-1(2k-1) is a perfect number and every even perfect number has this form. I will include a proof by Chris Caldwell. It uses the sigma function which is a function that finds the sum of the divisors. I will give link for the sigma function for those who are interested.
Proof: Suppose first that p = 2k-1 is a prime number, and set n = 2k-1(2k -1). To show n is perfect we need only show sigma(n) = 2n. Since sigma is multiplicative and sigma(p) = p+1 = 2k, we know
sigma(n) = sigma(2k-1)(sigma(p)) = (2k-1)2k = 2n.
This shows that n is a perfect number.
On the other hand, suppose n is any even perfect number and write n as 2k-1m where m is an odd integer and k>2. Again sigma is multiplicative so
sigma(2k-1m) = sigma(2k-1)(sigma(m)) = (2k-1)(sigma(m)).
Since n is perfect we also know that
sigma(n) = 2n = 2km.
Together these two criteria give
2km = (2k-1)(sigma(m)),
so 2k-1 divides 2km hence 2k-1 divides m, say m = (2k-1)M. Now substitute this back into the equation above and divide by 2k-1to get 2kM = sigma(m). Since m and M are both divisors of m we know that
2kM = sigma(m) > m + M = 2kM,
so sigma(m) = m + M. This means that m is prime and its only two divisors are itself (m) and one (M). Thus m = 2k-1 is a prime and we have prove that the number n has the prescribed form.
It is even easier to prove that if for some positive integer n, 2n-1 is prime, then so is n.
So this question about perfect numbers has generated some interesting discussion of Mersenne primes and included some exciting number theory. This is a beautiful topic in math that is often not covered.
There are infinitely many perfect numbers so they cannot all be listed.
There is a one-to-one relationship between even perfect numbers and Mersenne primes. It is unknown whether there are any odd perfect numbers.
Yes. The next perfect numbers are 496 and 8128.
There are two perfect numbers, 6 and 28, that are less than 100.
No, there are no known perfect numbers between 1 and 30. The only perfect numbers that have been discovered are 6, 28, 496, and 8128.
create a program that iterates until it finds a perfect number, then store that perfect number into an array. Continue iterating until you find three more. Then, you have an array of four perfect numbers.
There are no perfect numbers between 20 and 30. Perfect numbers are numbers that are equal to the sum of their proper divisors, excluding the number itself. The perfect numbers within this range would be 28, but that is incorrect as 28 is not a perfect number.
6 and 28 are perfect numbers.
There are 48 different numbers that are considered to be perfect numbers. The perfect numbers that are up to 100 include 6 and 28.
No. The only perfect numbers less than 100 are 6 and 28. All known perfect numbers are even - it is unknown whether there are odd perfect numbers.
81. They are the perfect squares of numbers starting from 5.81. They are the perfect squares of numbers starting from 5.81. They are the perfect squares of numbers starting from 5.81. They are the perfect squares of numbers starting from 5.
By definition, ALL perfect squares are whole numbers!
6, 28, 496 and 8128 are the first four Perfect numbers.
Natural numbers which are the scales of some natural numbers are perfect squares
No. The first two "perfect numbers" are 6 and 28.
There are infinitely many perfect numbers so they cannot all be listed.
Other than what? The first perfect numbers are 6 and 28.