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### List of Perfect Numbers

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Date: 24 Mar 1995 03:37:47 -0500
From: Jirachai Chuchat
Subject: Perfect Numbers

Do you know any Perfect Numbers besides, 6, 28, and 496.  Do you know
where I can get a list of Perfect Numbers

Thanks.
Chris
```

```
Date: 24 Mar 1995 21:13:05 -0500
From: Dr. Sydney
Subject: Re: Perfect Numbers

Thanks for writing Dr. Math!  Perfect numbers are pretty interesting!
It turns out that perfect numbers are involved in several unsolved
problems of mathematics.  Euclid showed that the number

(*)   n = [2^(m-1)](2^m - 1)

is perfect if 2^m - 1 is a prime number.

Prime numbers of the form 2^m - 1 are called Mersenne primes, and it
is not known whether there are a finite or infinite number of Mersenne
primes.

We also don't 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
10^50. We do know that all even perfect numbers end in 6 or 8.

You wanted a list of perfect numbers. Well, as of 1990 (that is the
date of my source), there were 30 known perfect numbers, beginning
with 6, 28, 496, 8128, 33550336, and so on. The numbers get huge, so
it is easier to write them in the notation (*) above. The 30 known
perfect numbers correspond to the following values of m: 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,
132049, and 216091.

Some books you might want to look up on perfect numbers are:

Dickson, L.E.: _History of the Theory of Numbers_, 1, Chelsea,
reprint, 1952.

Nankar, M.L.: "History of perfect numbers," Ganita Bharati 1,
no. 1-2 (1979), 7-8.

Hagis, P.: "A Lower Bound for the set of odd Perfect Prime Numbers",
Math. Comp. 27, (1973), 951-953.

Riele, H.J.J. "Perfect Numbers and Aliquot Sequences" in H.W. Lenstra
and R. Tijdeman (eds.): _Computational Methods in Number Theory_,
Vol. 154, Amsterdam, 1982, pp. 141-157.

Riesel, H. _Prime Numbers and Computer Methods for Factorisation_,
Birkhauser, 1985.

Hope this helps!  Write back with any more problems.

--Sydney "Dr. Math"
```

```
Date: 24 Mar 1995 21:24:49 -0500
From: Christopher Yeung
Subject: Re: Perfect Numbers

What are the perfect numbers "so on"?  I found that 8192 is not a
Perfect Number. Thanks.
```

```
Date: 24 Mar 1995 22:52:46 -0500
From: Dr. Ken
Subject: Re: Perfect Numbers

Yup, you're right, 8192 is not a perfect number.

But I don't think we ever said it was.  If you plug in m=7 to the
formula n = 2^(m-1) * (2^m - 1), you don't get 8192, you get 8128.
Is this what happened?

You asked for the numbers "and so on."

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

-Ken "Dr." Math
```
Associated Topics:
College Number Theory

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