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a_1 is even if prime factorisation of perfect n>6, p_1^{a_1}p_2^{a_2} ....

Source: Switzerland - 2022 Swiss Final Round p7

November 17, 2022

number theoryEvenprime factorizationPerfect Numbersperfect number

Problem Statement

Let

n > 6

be a perfect number. Let

p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}

be the prime factorisation of

n

, where we assume that

p_1 < p_2 <...< p_k

and

a_i > 0

for all

i = 1,...,k

. Prove that

a_1

is even.
Remark: An integer

n \ge 2

is called a perfect number if the sum of its positive divisors, excluding

n

itself, is equal to

n

. For example,

6

is perfect, as its positive divisors are

\{1, 2, 3, 6\}

and

1+2+3=6