Perfect Numbers?

Question

What is the largest perfect number? What mathematician made an attempt at deriving a formula to produce more of them?

Answer 1

There is no largest perfect number, but they are hard to find, at least after the first few, without a super computer to crunch numbers for you... Usually they find a Mersenne Prime and for each of those there is an associated Perfect number...



I picked this off the St Andrews site, and it is dated 2001 so it may well not be current,,, but will give you an idea..
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At the moment the largest known Mersenne prime is 213466917 - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 213466916(213466917 - 1). It was discovered in December 2001 and this, the 39th such prime to be discovered, contains more than 4 million digits.
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and at the Math forum I found a list of the first few, you can see they get big in a hurry:
Here are the first few perfect numbers:

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 others at

Known Perfect Numbers
http://amicable.homepage.dk/perfect.htm


A good site to understand a little more about these ideas and their history is at http://primes.utm.edu/mersenne/index.html

Hope that helps...

Answer 2

A perfect number is one that is the sum of its own factors (excluding itself).

The first four perfect numbers were known to the ancient Greeks:

6 = 1 + 2 + 3

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

As you will notice, each is constructed of powers of 2 and one less than a power of 2. generated by a prime number (3, 7, 31 and 127 respectively.

There proves to be a 1-to-1 relationship between Mersenne Primes and Perfect Numbers. It is known there is no largest prime number and therefore no largest Mersenne prime number, We now know of 44 Mersenne primes (the last ten unearthed by the collaborative efforts of G.I.M.P.S., the Great Internet Mersenne Primes Search) and therefore of 44 Perfect Numbers,

Answer 3

In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n.

The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128

Leonhard Euler proved that the formula 2^(n−1)(2^n − 1) will yield all the even perfect numbers.\

The largest known such perfect number todate is 2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits.

It is unknown whether there are any odd perfect numbers.

Answer 4

23 that's one hell of a number do some research on it see the movie its a crazy number