杨辉三角与素数的关系

Relationship between Yang Hui Triangle and Prime Number

设p为素数,s,t∈N,a=sum form i=0 to ∞(aipi),r=sum form i=0 to ∞(ripi),这里ai,ri∈N,0≤ai≤p-1,0≤i≤t,0≤ri≤p-1,0≤i≤s,证明了Car≡Ca0r0…Casrs(mod p)和C(a+r)r≡C(a0+r0)r0C(a1+r1)r1…C(a1+r1)r1(mod p)两个同余式.据此导出了杨辉三角的第a行以及第0行至第a行的二项系数中,使Car■0(mod p)的个数和使Car≡0(mod p)的个数,推出了斜列{C(a+r)r:r=0,1,…}中使C(a+r)r■0(mod p)的个数和使C(a+r)r≡0(mod p)的个数.

p is a prime number, and s,t∈N,a=t∑i=0 aip^i,r=s∑i=0 rip^i where ai,ri∈N,0≤ai≤p-1,0≤i≤t,0≤ri≤p-1,0≤i≤s, two congruence expression,Ca^r=Ca0^r0…Cas^rs（mod p） and Ca＋r^r≡Ca0＋r0^r0 Ca1＋r1^r1…Cat＋rt^rt（mod p） are proved in the paper, Further more, we deduces the number of binomial coefficients from 0 to a rows of Yang Hui Triangle which can make Ca^r≡0（mod p） and make Ca^r≡0（mod p） and also get the numbers which make Ca＋r^r≠0（mod p） and Ca＋r^r≡0（mod p） in oblique column {Ca＋r^r：r=0,1,…}