Odd Integers N with Five Distinct Prime Factors for which 2 - 10 -12 &lt; σ(N)/N &lt; 2 + 10 -12

Kishore, Masao

doi:10.2307/2006281

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Cited by 19 publications

(22 citation statements)

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“…(A), (B) and (C) (implicitly) were proved by Cattaneo (1951), (D) is due to Kishore (1978), and (E) and (F) were proved by Abbott and others (1973).…”

Section: Introductionmentioning

confidence: 75%

Some results concerning quasiperfect numbers

Hagis

Cohen²

1982

J Aust Math Soc A

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“…(A), (B) and (C) (implicitly) were proved by Cattaneo (1951), (D) is due to Kishore (1978), and (E) and (F) were proved by Abbott and others (1973).…”

Section: Introductionmentioning

confidence: 75%

Some results concerning quasiperfect numbers

Hagis

Cohen²

1982

J Aust Math Soc A

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“…Moreover, the range of the inequality for |σ (N)/N − 2| can be made very small even when N has a few prime factors. Examples of odd integers with only five distinct prime factors have been found, which produce a ratio nearly equal to 2: |σ (N)/N − 2| < 10 −12 [28]. Since it becomes progressively more difficult to establish the inequalities as the number of prime factors increases, a proof by method of induction based on this algorithm cannot be easily constructed.…”

Section: Introductionmentioning

confidence: 99%

A rationality condition for the existence of odd perfect numbers

Davis

2003

International Journal of Mathematics and Mathematical Sciences

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A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ (N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10 300 . The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and it is shown that the arithmetic primitive factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation. This equation is one example of a more general prime equation, (q n j − 1)/(q n i − 1) = p h , and the demonstration of the nonexistence of solutions when h ≥ 2 requires the proof of a special case of Catalan's conjecture. General theorems on the nonexistence of prime divisors satisfying the rationality condition and odd perfect numbers N subject to a condition on the repunits in factorization of σ (N) are proven.

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“…In this paper using the technique of [4], we prove Theorem. // jV is OT, N has at least ten distinct prime factors.…”

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confidence: 98%