Stephen P. Richards's A Number for Your Thoughts PDF

By Stephen P. Richards

ISBN-10: 0960822402

ISBN-13: 9780960822409

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It is called the integral logarithm and is usually written as Li (n). For those readers who know a little calculus (and the rest of you need not worry at all about it) the formal definition of this function is n Li(n) = I2 dx/ln(x). The following table indicates the accuracy of Li (n) when compared with the actual count of primes 1r(n) less than or equal to n: n 1r(n) Li(n) 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 9,592 78,498 664,579 5,761,455 50,847,534 9,630 78,628 664,918 5,762,209 50,849,235 More sophisticated functions can do even better but for us Li (n) is quite sufficient and, if it were plotted in Figure 2, the difference between it and the real 1r(n) could not be seen on the scale of that figure.

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 is now known that there are others as well, but they are of the utmost rarity. In checking numbers of up to one thousand repeating ones, only two others have been located. They have respectively 23 and 317 ones. Needless to say, it is not known whether prime numbers of this form exist to arbitrarily large values. 42 A NUMBER FOR YOUR THOUGHTS Related to the repunit prime numbers are those which are all of this kind except for the first or last digit. Some examples of prime numbers of this form are 111,111,113 11,111,111,113 11,111,117 11,111,119 the last two making a rather remarkable pair of prime twins.

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A Number for Your Thoughts by Stephen P. Richards


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