what is a perfect number?

Answer 1

Perfect numbers are the sum of their own factors and the last answerer has demonstrated that, helpfully

But a related phenomenon are amicable numbers, which are the sum of each other's factors, and sociable numbers in which a is the sum of b's factors, b is the sum of c's factors, c is the sum of d's factors and d is the sum of a's factors,

The lowest pair of amicable numbers is 220 and 284, These were known to the Ancient Greeks

220's factors:
110+55+44+22+20+11+10+5+4+2+1 = 284

284's factors:
142+71+4+2+1 = 220

A general formula by which these numbers could be derived was invented circa 850 by Thabit ibn Qurra (826-901): if

p = 3 × 2^(n-1) - 1,
q = 3 × 2^(n) - 1,
r = 9 × 2^(2n-1) - 1,
where n > 1 is an integer and p, q, and r are prime numbers, then 2^(n) pq and 2^(n) r are a pair of amicable numbers. This formula gives the amicable pair (220, 284)

n=2 p=5 q=11 r=71 so 4pq=220 and 4r=284

And it generates the pair (17296, 18416) and the pair (9363584, 9437056).

The pair (6232, 6368) are amicable, but they cannot be derived from this formula. In fact, this formula produces amicable numbers for n = 2, 4, and 7, but for no other values below 20000.

In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists.

Also, every known pair shares at least one common factor. It is not known whether a pair of coprime amicable numbers exist, though if they do, their product must be greater than 10^67

Amicable numbers have been studied by Al Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980-1037), René Descartes (1596-1650), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others. Thabit's formula was generalized by Euler.

In a rather Eurocentric account "It was not until 1636 that the great Pierre de Fermat discovered another pair of amicable numbers (17296, 18416). Later Descartes gave the third pair of amicable numbers i.e. (9363584, 9437056)."

In the 18th century great Euler drew up a list of 64 amiable pairs (two of which later shown to be unfriendly). B.N.I. Paganini, a 16-years-old Italian, startled the mathematical world in 1866 by announcing that the numbers 1184 and 1210 were friendly. It was the second lowest pair and had been completely overlooked until then, Even Euler's list of Amicable pairs does not contain it.

Today about 10,306,909 pairs of amicable numbers are known.

the next 12 smallest amicable pairs after (220, 284) are

Paganini 1184 & 1210
Euler 2620 & 2924
Euler 5020 & 5564
Euler 6232 & 6368
Euler 10744 & 10856
Brown (1939) 12285 & 14595
al-Banna 1300, Farisi 1300, Fermat 1636
17296 & 18416
Euler 63020 & 76084
Euler 66928 & 66992
Euler 67095 & 71145
Euler 69615 & 87633
Rolf (1964) 79750 & 88730

SOCIABLE NUMBERS

Each is the sum of the factors of the preceding one in the cycle. The cycles can be of different lengths.

1 12496 14288 15472 14536 14264 (cycle of 5)

2 14316 19116 31704 47616 83328 177792 295488 629072 589786 294896 358336 418904 366556 274924 275444 243760 376736 381028 285778 152990 122410 97946 48976 45946 22976 22744 19916 17716 (cycle of 28)

3 1264460 1547860 1727636 1305184 (cycle of 4)

Answer 2

Three known to the ancient Greeks were 6, 28, 496, but 8124 cannot be the 4th one

6 = 3 + 2 + 1
28 = 14 + 7 + 4 + 2 + 1
496 = 248 + 124 + 62 + 31 + 16 + 8 + 4 + 2 + 1
8124 does not = 4062 + 2708 + 2031 + 1354 + 677 as their sum = 10832

Answer 3

A perfect number is one which is the sum of all of its divisors excluding itself. For example, the divisors of 6 are 1,2, and 3 and 6=1+2+3.

We know the general form of all even perfect numbers. They are of the form
2^(n-1) [2^n -1] where 2^n -1 is prime.
So, for example, if n=5, 2^5 -1=31 is prime. So 2^4 [2^5 -1]=16*31=496 is perfect. Primes of the form 2^n -1 are called Mersenne primes. Most of the largest primes known are of this form. We do not know if there are infinitely many.

It is not known if there are any odd perfect numbers. This is perhaps the oldest open problem in mathematics.

Answer 4

A perfect number is defined as an integer which is the sum of its proper positive divisors.

e.g. 6 is a perfect number since its divisors, 1 2 and 3 when added together make 6

Answer 5

6

Answer 6

10

Answer 7

A perfect number is a number that is equal to the sum of its integral factors (for example 28, whos divisors are 1,2,4,7 and 14.

The first four PERFECT numbers are: 6, 28, 496 and 8128,

Answer 8

7

Answer 9

8

Answer 10

0 is a perfect number