问题: 数学竞赛的题目
有P个质数A1,A2,……AP，是等差数列，d>0,A1>P
求证:(1)当P为质数时,P|d(整除)
(2)当P=15,d>30000
解答:
1) Since A1>P, d>0 and Ai i=1,2,...,P are primes
P does not |Ai,
The remainders of Ai divided by P can only be among
1,2,...,P-1
So there are two Ai, Aj such that Ai=Aj(mod P)
P|(Ai-Aj), or P|(i-j)*d
|i-j| is less than P, hence P|d
2) P=15
By 1) All primes less than 15 should divids d:
2,3,5,7,11,13
2*3*5*7*11*13=30030
So d>=30030>30000
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