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K n of the form 1 + 2 + 4 + 8 + . . . + 2 n don'tneed to be


Academic year: 2022

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Texte intégral


Épreuve de setion européenne

Perfet numbers


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


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

double proportion,untilthesumofallbeomesprime,andifthesummultipliedinto


With theadvantages of modernnotation, we anexpresswhat Eulid meantmorepreisely:

ifwebeginwith 1andadd toitsuessivelyhigherpowersof2sotheresultingsum

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

is aprime number,thenthe number

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

, formed by


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

byitslast summand

2 n


Notein passingthat the numbers

K n

of the form

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

don'tneed to be

primeat all. Eulid'sperfetnumbertheorem appliedonlytothosespeialaseswherethissum




1 + 2 + 4 + 8 + 16 = 31

,aprime. Then,

N = 16 × 31 = 496

should beperfet. To

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


AdaptedfromJourney through genius, thegreattheoremsof mathematis,byWilliamDunham


1. Usethetexttoexplainthefollowingwords: primenumber,properdivisor,perfetnumber.

2. Chekthat28is perfet.

3. Find asimplerformulafor

K n


N n


4. In orderto hekthat numbers

K n

of theform

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

don't needto be

prime atall',ompleteatable ofvaluesof

K n




5. Chekthat

N 3


6. ProofofEulid'stheorem: Wesupposethat



(a) Listallthedivisorsof

2 n


(b) Henededuealltheproperdivisorsof

N n = 2 n × K n


K n = 2 n+1 − 1


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

N n

is perfet.


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