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By Stephen P. Richards
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Extra resources for A Number for Your Thoughts
Let us write down a few more examples: 6=2X3 30 = 2X3X5 210 = 2X3X5X7 625 = 5X5X5X5. This tells us immediately that p(6)=2, p(30)=3, p(21O)=4, and p(625)=1. What sort of function is this p(n)? It is evident from the few examples already given that it certainly does not increase steadily with increasing values of n. Moreover, since the prime numbers go on forever, and since p (n) is equal to 1 for all prime numbers, there must be some values of p (n) equal to 1 no matter how large n becomes. On the other hand, the value of p (n) for other numbers can grow much larger as n increases.
It follows that we have WHAT IS SO SPECIAL ABOUT 6174? 57 generated a loop and that therefore no unique self-generating number can possibly exist for 5-digit integers.
E) Finally, repeat the operations (b), (c), and (d) on the new number and continue. 54 A NUMBER FOR YOUR THOUGHTS The object of the exercise is to see what happens as you go on and on. Let us try a very simple example to begin with; say a 2-digit number like 90. For this particular case the above procedure takes the form: 90 81 63 72 54 90 - 09 18 36 27 45 09 = 81 = 63 = 27 = 45 = 09 = 81 and we notice that the last line is exactly the same as the first, so that we have formed a loop which regenerates itself over and over again.
A Number for Your Thoughts by Stephen P. Richards