91Ó°ÊÓ

A positive integer \(n\) is said to be a deficient number if \(\sigma(n)<2 n\) and an abundant number if \(\sigma(n)>2 n\). Prove each of the following: (a) There are infinitely many deficient numbers. [Hint: Consider the integers \(n=p^{k}\), where \(p\) is an odd prime and \(k \geq 1\).] (b) There are infinitely many even abundant numbers. [Hint: Consider the integers \(n=2^{k} \cdot 3\), where \(k>1 .\) ] (c) There are infinitely many odd abundant numbers. [Hint: Consider the integers \(n=945+k\), where \(k\) is any positive integer not divisible by \(2,3,5\), or 7 . Because \(945=3^{3} \cdot 5 \cdot 7\), it follows that \(\operatorname{gcd}(945, k)=1\) and so \(\sigma(n)=\sigma(945) \sigma(k) .1\)

Prove that any multiple of a perfect number is abundant.

For any Fermat number \(F_{n}=2^{2 \cdot}+1\) with \(n>0\), establish that \(F_{n} \equiv 5\) or \(8(\bmod 9)\) according as \(n\) is odd or even. [Hint: Use induction to show, first, that \(2^{2^{n}} \equiv 2^{2^{n-2}}(\bmod 9)\) for \(\left.n \geq 3 .\right]\)