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71, 701, 7001, 70001, and 700001 are all prime, but 7000001=197x35533. What’s going on here?
Recall that a whole number greater than 1 is prime if the only whole numbers dividing it are 1 and itself. These have played a central role in mathematics for at least a couple of thousand years, and the first few are easily seen to be 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on.
Answering even the simplest of questions about the sequence of prime numbers is infamously difficult. Are there infinitely many pairs of them that differ by just 2? Are there infinitely many whose digits are all equal to 1? Are there infinitely many that are 1 more than a power of 2?
But the titular sequence’s ability to give us primes is quite striking—can we explain what’s going on here?
Simple divisibility tests
If you know some of the well-known and easy-to-apply basic divisibility tests for small numbers, then you can immediately see that certain numbers completely avoid ever dividing any member of the sequence altogether.
- Since the last digit is never even, it follows that the prime 2 never divides these numbers.
- Since the last digit is never equal to 0 or 5, it follows that the prime 5 never divides these numbers.
- Since the sum of the digits…
