The second largest prime factor of an odd perfect number
نویسندگان
چکیده
منابع مشابه
The Second Largest Prime Factor of an Odd Perfect Number
Recently Hagis and McDaniel have studied the largest prime factor of an odd perfect number. Using their results, we begin the study here of the second largest prime factor. We show it is at least 139. We apply this result to show that any odd perfect number not divisible by eight distinct primes must be divisible by 5 or 7.
متن کاملThe second largest prime divisor of an odd perfect number exceeds ten thousand
Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions ...
متن کاملOn the Largest Prime Divisor of an Odd Perfect Number . II
It is proved here that every odd perfect number has a prime factor greater
متن کاملThe third largest prime divisor of an odd perfect number exceeds one hundred
Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. It is unknown whether or not odd perfect numbers exist, although many conditions necessary for their existence have been found. For example, Cohen and Hagis ha...
متن کاملOn the largest prime divisor of an odd harmonic number
A positive integer is called a (Ore’s) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than 1 is even. If Ore’s conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than 1 must be divisible by a prime greater than 105.
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1975
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1975-0371801-7