References: [ Home | Author index | Key index | Search ]

Item(s) in original BibTeX format

@article{IS2003,
	author={D. E. Iannucci and R. M. Sorli},
	title={On the total number of prime factors of an odd perfect number},
	abstract={We say $n\in$ {\bf N} is {\em perfect} if $\sigma(n)=2n$, where $\sigma(n)$
		denotes the sum of the positive divisors of $n$. No odd perfect numbers
		are known, but it is well known that if such a number exists, it must have
		prime factorization of the form $n=p^{\alpha}\prod_{j=1}^{k}q_j^{2\beta_j}$,
		where $p$, $q_1$, $\dots$, $q_k$ are distinct primes and $p\equiv\alpha\equiv1\pmod{4}$.
		We prove that if $\beta_j\equiv1\pmod{3}$ or $\beta_j\equiv2\pmod{5}$ for
		all $j$, $1\le j\le k$, then $3\nmid n$. We also prove as our main result
		that $\Omega(n)\ge37$, where $\Omega(n)=\alpha+2\sum_{j=1}^{k}\beta_j$.
		This improves a result of Sayers ($\Omega(n)\ge 29$) given in 1986. },
	journal= mc,
	year= 2003,
	volume= 72,
	number=244,
	pages={2077--2084},
	mrnumber={1986824},
	address={\url{http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-03-01522-9}}
}