[精选数学]Adding-distinct-congruence-classesVIP专享VIP免费

A d d i n g d i s t i n c t c o n g r u e n c e c l a s s e sm o d u l o a p r i m eN o g a A l o n∗Me l v y nB . N a t h a n s o n†I m r e R u z s a‡1T heE r d˝o s -He i l br o nn co nj e ct ur eT h eC au c h y - D av e n p or t t h e or e ms t at e s t h at i f A an dBar en on e mp t ys e t s ofc on gr u e n c ec l as s e smod u l o a p r i mep , an di f | A| =kan d| B | =l , t h e nt h es u ms e t A +Bc on t ai n s at l e as t mi n ( p , k +l −1) c on gr u e n c e c l as s e s . I t f ol l ow st h at t h e s u ms e t 2A c on t ai n s at l e as t mi n ( p , 2k −1) c on gr u e n c e c l as s e s .E r d ˝osan dHe i l b r on nc on j e c t u r e d30 y e ar s ago t h at t h e r ear eat l e as t mi n ( p , 2k −3)c on gr u e n c ec l as s e s t h at c anb ew r i t t e nas t h es u mof t w o d i s t i n ct e l e me n t s ofA. E r d ˝os h as f r e q u e n t l yme n t i on e dt h i s p r ob l e mi nh i s l e c t u r e s an dp ap e r s ( f ore x amp l e , E r d ˝os - G r ah am[ 4, p .95] ) .T h ec on j e c t u r ew asr e c e n t l yp r ov e nb yD i as d a S i l v a an dHami d ou n e[ 3] , u s i n g l i n e ar al ge b r a an dt h er e p r e s e n t at i ont h e or yof t h es y mme t r i cgr ou p .T h ep u r p os eof t h i s p ap e r i s t o gi v ea s i mp l ep r oof of t h eE r d ˝os - He i l b r on nc on j e c t u r et h atu s e son l yt h emos te l e me n t ar yp r op e r t i e s of p ol...