By Stephen P. Richards
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Extra info for A Number for Your Thoughts
At any finite 'distance post', on the other hand, the race does have a leader (or is tied) and the competition is keen. It happens that this particular race is rather unusual and has been pursued all the way up to prime numbers of the order 2XI0 1O• Those interested in the details will find them in Appendix 2 at the back of the book. The 'groups of four' number race set out above is not the only race in the prime number stakes by any means. For example, all the primes larger than 5 can be grouped by sixes.
Are absolute, while the latter depend on the counting base. Nevertheless, as we have seen with Benford in the previous chapter, a lot of fun can be had with representations as well as absolutes, and in this (and several of the following) chapters we shall look at the decimal representation in some detail. Whether as decimal representations, or as numbers in general, 495 and 6174 certainly do not exhibit any outward appearance of undue mystique; one is odd and one is even; they are not prime or perfect.
In other words, there can exist arbitrarily long sequences of numbers all of which are composite. This is proved by noting that when we have reached a number of the form lX2X3X4X5X·· . Xn, which mathematicians write in the shorthand notation n! +1 are all composite. For example, n! + 2 must be divisible by 2 (since n! by definition is divisible by 2) and n! +n which is divisible by n. Since we can choose n to be as large as we please, we can generate as many consecutive composites as we like. It follows that when we get up to extremely large numbers, the next prime can be as close as the next odd number or can be a very long way away indeed.
A Number for Your Thoughts by Stephen P. Richards