The Abundancy Index of Divisors of Odd Perfect Numbers
نویسنده
چکیده
If N = qkn2 is an odd perfect number, where q is the Euler prime, then we show that σ(n) ≤ qk is necessary and sufficient for Sorli’s conjecture that k = νq(N) = 1 to hold. It follows that, if k = 1 then the Euler prime q is the largest prime factor of N and that q > 10500. We also prove that qk < 23n 2.
منابع مشابه
Does Ten Have a Friend?
Any positive integer n other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of n. 1 The Abundancy Index For a positive integer n, the sum of the positive divisors of n is d...
متن کاملAbundancy “ Outlaws ” of the Form σ ( N ) +
The abundancy index of a positive integer n is defined to be the rational number I(n) = σ(n)/n, where σ is the sum of divisors function σ(n) = ∑ d|n d. An abundancy outlaw is a rational number greater than 1 that fails to be in the image of of the map I. In this paper, we consider rational numbers of the form (σ(N) + t)/N and prove that under certain conditions such rationals are abundancy outl...
متن کاملOdd perfect numbers have at least nine distinct prime factors
An odd perfect number, N , is shown to have at least nine distinct prime factors. If 3 N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
متن کاملRFSC 04-01 Revised A PROOF OF THE ODD PERFECT NUMBER CONJECTURE
It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k+1) ∏l i=1 q 2αi i to establish that there do not exist any odd integers with equality between σ(N) and 2N. The existence of distinct prime divisors in the repunits in σ(N) follows from a theorem on the primitive divisors of the ...
متن کاملThe Imperfect Fibonacci and Lucas Numbers
A perfect number is any positive integer that is equal to the sum of its proper divisors. Several years ago, F. Luca showed that the Fibonacci and Lucas numbers contain no perfect numbers. In this paper, we alter the argument given by Luca for the nonexistence of both odd perfect Fibonacci and Lucas numbers, by making use of an 1888 result of C. Servais. We also provide a brief historical accou...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1103.1090 شماره
صفحات -
تاریخ انتشار 2011