I am puzzled, though, by its treatment of Euclid’s famous proof of the infinite order of the set of primes.
Not because it gets the proof wrong – but because I do not understand the answer it gives to one part of one exercise.
Now the proof (this is my summary not the book’s) runs like this:
Suppose you decide that the number of primes is finite, the set (where is 2 and so on). You then form a number from the product of these primes plus 1 – . This number cannot be a prime and yet is not divisible by any of the primes. Hence we have a contradiction and the set of primes must be infinite.
Now the book asks the following:
The proof of [the theorem] gives a method for finding a prime number from any in a given list of prime numbers:
(a) Use this method to find a prime different from 2, 3, 5 and 7
(b) Use this method to find a prime different from 2, 5 and 11
Now, the first seems easy enough:
and the book agrees with my calculation.
But what of the second?
Well, , but that’s not a prime at all. The book gives the answer as “3 and 37” (which obviously are primes) but I cannot see what formal method is used from the list of “2, 5 and 11” to generate these numbers.
The obvious suggestion is that I have missed the “method” in the theorem as set out in the book, but I have reread it many times and I cannot see where I have missed anything (it is really just a longer version of my summary).
Can someone set out the formal method that would work with a partial list such as 2, 5 and 11?
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- Prime Numbers: The Most Mysterious Figures in Math (mathematicslibrary.wordpress.com)
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- The First Six Books of The Elements of Euclid (ministryoftype.co.uk)
- For every natural number N, there’s a Cantor Crank C(n) (scientopia.org)
- A Presidential Pythagorean Proof (blogs.scientificamerican.com)
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