Find the minimum common difference for an increasing sequence of 6 prime numbers
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The minimum common difference must be a multiple of $30$, because if the difference is odd then every other number will be even, and if the difference is not a multiple of $3$ then two of the numbers will be a multiple of $3$, and one of the numbers will be a multiple of $5$ if the common difference is not a multiple of $5$.
$7,37,67,97,127,157$ has difference $30$.