形如1/2(11^(2^n)+1)的孤立数

Anti-Sociable Numbers with the Form 1/2(11^(2^n)+1)

孤立数一直是数论研究的一个重要课题。最近,在孤立数研究方面取得了一些进展。2006年,沈忠华证明了1/2(5^(2^n)+1)都是孤立数;2007年,蒋自国、曹型兵证明了1/2(3^(2^n)+1)都是孤立数;2011年,张四保、吕明富证明了1/2(7^(2^n)+1)都是孤立数;2012年,管训贵证明了1/3(2~p+1)都是孤立数。本文运用初等数论的方法证明了:1/2(11^(2^n)+1)都是孤立数,这里n是任意的正整数。

Anti-sociable numbers are important topics in Number Theory, and some development has been achieved in the past years. In 2006, Shen Zhonghua proved 1/2（5^2n＋1） are anti-sociable numbers; In 2007, Jiang Ziguo and Coa Xingbing proved 1/2（5^2n＋1） are anti-sociable numbers; In 2011, Zhang Sibao and Lv Mingfu proved 1/2（7^2n＋1） are anti-sociable numbers; In 2012, Guan Xungui proved 1/3（2^p＋1） are anti-sociable numbers. In this paper, some elementary number theory methods are used to prove that 1/2（11^2n＋1） are anti-sociable numbers, where n is a positive integer.