Abstract
Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form <TEX>$2^{\alpha}p_1p_2$</TEX> and <TEX>$2^{\alpha}p_1^2p_2$</TEX>, 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 <TEX>$p_1=2^{{\alpha}+1}-1$</TEX> and <TEX>$p_2={\frac{p^2_1+p_1+1}{3}}$</TEX>, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.
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