Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A CLASS OF NEW NEAR-PERFECT NUMBERS

  • LI, YANBIN (Institute of Mathematics and Software Science Sichuan Normal University) ;
  • LIAO, QUNYING (Institute of Mathematics and Software Science Sichuan Normal University)
  • Received : 2014.08.19
  • Published : 2015.06.01
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Abstract

Let ${\alpha}$ be a positive integer, and let $p_1$, $p_2$ be two distinct prime numbers with $p_1$ < $p_2$. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form $2^{\alpha}p_1p_2$ and $2^{\alpha}p_1^2p_2$, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form $p_1=2^{{\alpha}+1}-1$ and $p_2={\frac{p^2_1+p_1+1}{3}}$, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.

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References

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