Question

Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number.

Answer 1

The statement that "σ(n) < 2n holds true for all n of the form n = p²" has been proved.

Let p be any prime number, and let σ(n) be the sum of all positive divisors of the integer n.

As p is a prime number, and 2 is the smallest prime number, so, p$\geq$2

So, the positive divisors of the integer n are: 1,p,p².

As σ(n) represents the sum of all positive divisors of the integer n.

σ(n)=1+p+p²

In order to prove that σ(n) < 2n,for all n of the form n = p².

1+p+p²<2p²

p²-p-1>0

It is know that, p$\geq$2.

So, p²-p-1$\geq$1

Thus, σ(n) < 2n holds true for all n of the form n = p².

Learn more about prime numbers here:

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