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Épreuve de setion européenne

Perfet numbers

Aperfetnumberisanaturalnumberthatisthesumofitsproperdivisors(alldivisorsexept

thenumberitself). Forexample,28is perfet. InTheElements,bookIX,proposition36,Eulid

wrote:

If asmanynumbersas weplease, beginningfrom a unit,are set outontinuously in

double proportion,untilthesumofallbeomesprime,andifthesummultipliedinto

With theadvantages of modernnotation, we anexpresswhat Eulid meantmorepreisely:

1 + 2 + 4 + 8 + . . . + 2 n

is aprime number,thenthe number

N n = 2 n (1 + 2 + 4 + 8 + . . . + 2 n )

, formed by

multiplyingthesum

1 + 2 + 4 + 8 + . . . + 2 n

byitslast summand

2 n

,mustbeperfet.

Notein passingthat the numbers

of the form

1 + 2 + 4 + 8 + . . . + 2 n

don'tneed to be

primeat all. Eulid'sperfetnumbertheorem appliedonlytothosespeialaseswherethissum

indeedturnsouttobeaprime.

Forinstane,

,aprime. Then,

N = 16 × 31 = 496

should beperfet. To

seethat itis, welistalltheproperdivisorsof496namely, 1,2,4,8,16, 31,62, 124and248

Questions

2. Chekthat28is perfet.

3. Find asimplerformulafor

K n

anddeduethegeneralexpressionof

N n

.

4. In orderto hekthat numbers

of theform

1 + 2 + 4 + 8 + . . . + 2 n

don't needto be

prime atall',ompleteatable ofvaluesof

,for

rangingfrom1to5.

5. Chekthat

N 3

isnotperfet.

6. ProofofEulid'stheorem: Wesupposethat

n

isanaturalnumber.

(a) Listallthedivisorsof

2 n

.

(b) Henededuealltheproperdivisorsof

,where

K n = 2 n+1 − 1

isprime.

() Add up all the properdivisors found in the previous question and provethat

N n

is perfet.

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