Odd square-free numbers and perfect numbers.
Number Theory
Level
pending
How many odd square-free numbers are perfect numbers.
Square-free numbers are integers that are not divisible by any perfect square.
Perfect numbers are integers that are half the sum of their divisors.
1
2
Infinitely Many
0
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1 solution
Assume N is such a number and that N=pn where p is an odd prime number. Because N is square-free, gcd(p,n)=1. If the sum of the divisors of N is S and the sum of the divisors of n is s, then (p+1)s=S. Because we assumed N is a perfect number, (p+1)s=2N. Because N is odd and, since n is square-free, s is even, 2N/s is odd, meaning p+1 is odd and thus p is even, a contradiction. Therefore, no odd square-free number is also a perfect number. So the answer is 0.