Twenty-one prime numbers are in arithmetic sequence with difference d. Prove that d is divisible by 9699690.

Let p1, p2, ..., p21 be the prime numbers. Then, p1+d=p2, p2+d=p3, ..., p20+d=p21. Take any prime p<10. If d is not divisible by p, then at least 2 of the prime numbers are divisible by p. Since p is the only prime divisible by p, that is impossible. Therefore, d is divisible by p. Then, d is divisible by 2*3*5*7=210. Take any prime 10<p<20. If d is not divisible by p, then at least 1 of the prime numbers is divisible by p. Since p is the only prime divisible by p, pn=p for some n. Since d is divisible by 210, p2>210, ..., p21>210. Since p<20, p2>p, ..., p21>p. Therefore, p1=p. Then, p1+pd is one of the primes. That is impossible. Therefore, d is divisible by p. Then, d is divisible by 2*3*5*7*11*13*17*19=9699690.

Posted by Math Man on 2016-09-05 21:00:42