A Number for Your Thoughts by Stephen P. Richards

By Stephen P. Richards

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They must therefore all have a sum of digits equal to 1+ 2 + 3 + . + 8 + 9 = 45 which is exactly divisible by 9. Now there is a simple law of arithmetic which states that any number with a sum of digits which is divisible by 9 is itself divisible by 9. It follows that all ten-digit numbers of this kind are divisible by 9. None of them is therefore prime, emirp or otherwise. This means that the series above, which starts out so confidently with 4, 11, 42, 193, ... must come all the way back down to zero by the tenth term.

Thus, if we are purists and insist that a true emirp must generate a different prime number when reversed, then there is only one known 'cyclic emirp', namely 193,939. We may now speculate whether it is unique among all decimal numbers, or whether out there in the countless number of primes lies another waiting to be discovered. THE PRIME PRIMES Consider for a moment the prime number 73,939,133. There is something quite unusual about it. By dropping successively the rightmost digit, the remaining numbers are still all primes.

As we can see, both horses in that race experienced just such bursts of speed early in the race; at 11, 17, 23, 29 and at 41, 47, 53, 59 for the' 11' row. and at 61, 67, 73, 79 for the '7' row. In order to get truly impressive bursts of sustained speed it is unfortunately necessary to form sequences in which the constant number to be added is quite large. If this number is 210, for example, then the smallest prime number not dividing 210 exactly is 11, so that we might expect 10 consecutive primes of this kind to arise from time to time.

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