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

Well, d must be divisible by 2, in order that none of the numbers are divisible by 2. Similarly, d must be divisible by 3, in order that none of the numbers are divisible by 3. Same for 5, 7, 11, 13, 17 and 19.

Thus, d is divisible by 2*3*5*7*11*13*17*19 = 9699690.

(As noted in relation to another puzzle recently, this proof is complete, with no need to find 21 numbers prime numbers in arithmetic sequence)