关于Giuga猜想的一点探索

An Exploration on Giuga's Conjecture

证明以下两个结论 :(1)当 2 |r时 ,居加数n若存在 ,则必有n≡ 1(mod6 ) ;(2 )若合数n =p1p2 …pr 满足条件r(pr -1r - (pr- 1) r-1+(p2 - 1)… (pr- 1) ) <n +1,则n不满足 1n -1+2 n -1+… +(n - 1) n -1≡ - 1(modn)。这在某种程度表明居加猜想的正确性。

Two conclusions are proved:(1)If there exists a Giuga number N with even distinct prime divisors,we have n≡1(mod6);(2)A composite number N doesn't satisfy the condition1 n-1 +2 n-1 +...+(n-1) n-1 ≡-1(modn)if N satisfies the condition r(p r-1 r-(p r-1) r-1 +(p 2-1)...(p r-1))<n+1.This indicates the correctness of Giuga's conjecture in a sense.