设PQ都是正整数,且p/q=1-1/2+13-14+15-16...-1+设p,q都是正整数且p/q=1-1/2+1/3-1/4+1/5.-1/1318+1/1319求证1979|p

Question

设P Q都是正整数,且p/q=1-1/2+13-14+15-16...-1+
设p,q都是正整数且p/q=1-1/2+1/3-1/4+1/5.-1/1318+1/1319求证1979|p

Answer 1

最后是不是要问p能被1979整除啊?
S=1-1/2+1/3-1/4+1/5-...-1/1318+1/1319
=(1+1/2+1/3+1/4+...+1/1319)-2*(1/2+1/4+...+1/1318)
=(1+1/2+1/3+1/4+...+1/1319)-(1+1/2+...+1/659)
=1/660+1/661+...+1/1319
1/660+1/1319=1979/(660*1319),
1/661+1/1318=1979/(661*1318),
.
1/989+1/990=1979/(989*990).
相加,并记B=660*661*...*1318*1319,得
S=1979/(660*1319)+1979/(661*1318)+...+1979/(989*990)
=1979A/B,
1979是质数,大于B中所有质因数,1979A/B约简时1979不可能被约去,所以若S约简成p/q
则p一定能被1979整除.