By Goto, Kida
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Extra resources for Direct-interaction approximation and Renolds-number reversed expansion for a dynamical system
One of the topics Gauss examined in that book is the idea of finite arithmetic. You get a finite arithmetic whenever you use a counting system that periodically cycles back on itself and starts again. For instance, when you tell the time, you count the hours 1, 2, 3, and so on, but when you reach 12 you start over again, 1, 2, 3, and so on. Similarly, you count minutes from 1 to 60 and then start over again. This use of finite arithmetic to tell the time is why it is sometimes referred to as 'clock arithmetic'.
For example, you cannot divide 5 by 6. One way to see this is to observe that if you multiply any of the numbers from 1 to 11 by 6, the result will be an even number, and hence cannot be congruent to 5 modulo 12. However, in the case where the modulus n is a prime number, division is always possible. So, for a prime modulus, the corresponding modular arithmetic has all the familiar properties of ordinary arithmetic performed with the rational or the real numbers; in mathematician's language, it is a field.
So, if P is not prime, it must be evenly divisible by some prime different from (and hence bigger than) all of p1, . . , pn. In particular, there must be a prime bigger than all of p1, . . , pn, so again, the sequence can be continued. It is interesting to observe that, when you look at the number P = p1 × p2 × . . × pn + 1 used in Euclid's proof, you don't actually know whether P is itself prime or not. The proof uses two arguments, one that works when P is prime, one that works when it is not.
Direct-interaction approximation and Renolds-number reversed expansion for a dynamical system by Goto, Kida