Prove:If a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n} is a perfect number, then
2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4 ;
if moreover, a is odd, then the upper bound 4 may be reduced to 232. inequalitiesnumber theory proposednumber theory