Perfect Numbers 6 = Another example of a perfect number is 28; and we have 28 =

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1 What is a erfect umber? Perfect Numbers A erfect umber is a umber which equals the sum of its ositive roer divisors. A examle of a erfect umber is 6. The ositive divisors of 6 are 1,, 3, ad 6. The roer divisors of 6 are 1,, ad 3. Thus the umber 6 is cosidered a imroer divisor of itself. We have 6 = Aother examle of a erfect umber is 8; ad we have 8 = Perfect umbers were robably first itroduced by the Pythagoreas (for mystical reasos). How to fid erfect umbers? The first to give a aswer about this questio was Euclid. Euclid showed that if is a rime umber such that 1 is also a rime umber, the 1 the umber ( 1) is a erfect umber. This was stated as Proositio 36 of Book IX of Euclid's book "The Elemets", writte about 300 BC. Note that the erfect umbers give by Euclid are eve umbers. About two thousads years later, Euler showed that ay eve erfect umber has to have the form give by Euclid. I other words, the eve erfect umbers are described as follows: Euclid-Euler Theorem Ay eve erfect umber N is of the form 1 N = ( 1), where ad 1 are rime umbers. With this formula, we ca roduce eve erfect umbers by tryig rime umbers that make 1 rime. 1 1 ( 1) 3, rime 6 3 7, rime , rime , rime 818 1

2 11 07, ot rime , rime 33, 550, , rime 8, 589, 869, 056 Oe ca see that fidig a eve erfect umber deeds o whether the umber 1 is rime or ot. Prime umbers of the form 1 are called Mersee rimes ad are deoted by M : M = 1. Mari Mersee ( ), a Frech mok ad a amateur mathematicia, claimed, without roof, that M is rime whe =,3,5, 7,13,17,19,31, 67,17, 57 ad comosite for all rimes < 57. Later o, it was show that his claim is false for = 67, 57 ad that he missed the rimes = 61,89,107 from his list. I additio to roducig eve erfect umbers, Mersee rimes are ow cosidered a mai suly of large rimes. If the rime is very large, M becomes very large too. How ca oe check if M is rime. The commo test used to check rimality of M is due to Edouard Lucas ( ). A imroved versio of the test was give later by Derrick Lehmer ( ). This test ca be stated as follows: Lucas-Lehmer Test: Let { L k } be the sequece defied recursively as follows: L 1 =, Lk + 1 = Lk for k 1. The, for >, M is rime if ad oly if M divides L 1. Let us illustrate Lucas-Lehmer test by two examles. First we have { L k } = {,1,19,3763, }. As M 3 = 7 divides L = 1 ad M 5 = 31 divides L = 3763, the both M 3 ad M 5 are rimes. Nowadays, Luca-Lehmer Test is used by suercomuters to discover larger ad larger Mersee rimes. U to the time of writig this article, the largest kow Mersee rime is 3,58, It was discovered i Setember of 006. It has 9,808,358 decimal digits. We state ow some roerties of eve erfect umbers. The uits digit of a eve erfect umber is either 6 or 8. I case the uits digit is 8, the i fact it must eds with the digits 8. Every eve erfect umber, excet the first, leaves a remaider of 1 whe divided by 9 ad leaves a

3 remaider of whe divided by 6. Every eve erfect umber, excet the first, ca be exressed as a sum of cosecutive odd cubes; for examle, = 1 + 3, = How may eve erfect umbers are there? Accordig to the secial form of a eve erfect umber, the questio is equivalet to the followig questio: How may Mersee rimes are there? The exact aswer is ot kow; although evidece suggests that maybe there are ifiitely may Mersee rimes. U to the writig of this article, there are kow Mersee rimes ad hece there are kow eve erfect umbers. The latest o Mersee rimes ca be foud i the web site: htt:// Odd Perfect Numbers Our discussio above was cocetrated o eve erfect umbers. What about odd erfect umbers? Is there ay examle of a odd erfect umber? This is maybe the oldest usolved roblem i Mathematics. U to ow, o sigle examle has bee foud for a odd erfect umber. What we kow is some roerties that a odd erfect umber, if it exists, must satisfy. We list some of these roerties: a A odd erfect umber has to be of the form M, where is rime, ad a leave a remaider of 1 whe divided by, ad M is a ositive iteger.(this result is due to Euler) A odd erfect umber must have at least eight distict rime factors. 300 A odd erfect umber must be larger tha 10. Amicable Numbers Related to erfect umbers are amicable umbers. Two ositive itegers are called amicable umbers or form a amicable air if each is equal to the sum of the roer divisors of the other. For examle the two umbers 0 ad 8 are amicable umbers sice 0 = ad 8 =

4 The Arab mathematicia Thabit be Korrah (ith cetury AD) formulated a rule to fid amicable umbers. This rule ca be stated as follows: Thabit's Rule for amicable airs For a atural umber, if are all rimes, the the air is a amicable air. = q = 3 1 r = ( q, r) For examle if we take =, the = 5, q = 11, ad r = 71 are all rime umbers ad hece we get the amicable umbers M = 1 = 0 ad N = q = 8 idicated above. If we take =, we get the amicable umbers M = 3 7 = 1796 ad N = 1151 = Euler ( ) gave a geeralizatio of Thabit's Rule which ca be stated as follows: Euler's Rule for amicable airs Let ad m be ositive itegers such that 1 m 1. If m m = ( + 1) 1 m q = ( + 1) 1 + m m r = ( + 1) 1 are all rimes, the the air ( q, r) is a amicable air. Note that if m = 1 i Euler's Rule, we get Thabit's Rule. Eve though there are rules to geerate amicable umbers, it is ot kow whether or ot there are ifiitely may amicable airs. Refereces: 1. Burto, David M., Elemetary Number theory, 6 th editio, McGraw- Hill, 007.

5 . Klee, V. ad Wago, S., Old ad New Usolved Problems i Plae Geometry ad Number Theory, MAA, New York, Ore, Oystei, Number Theory ad its History, Dover Publicatios Ic., New York, Ya, Sog Y., Number Theory for Comutig, d editio, Sriger, Berli, 00. Ibrahim Al-Rasasi Jue,