20201011, 08:40  #12  
Sep 2002
Database er0rr
3446_{10} Posts 
Quote:
Please detail what the errors are so that I can fix them. 

20201011, 22:26  #13  
Dec 2012
The Netherlands
2×727 Posts 
Quote:
p5 The proof that root 2 is irrational relies on the uniqueness of prime factorization. As this is a special property of the integers (not true in all number systems), I would at least mention it. p6 Ordinary sets are unordered and may not have repeating elements (this is so that they correspond with properties  if you select all objects satisfying a certain condition, you want what you get to be a set). Orderings and multisets can be constructed from ordinary sets if needed (in fact, so can everything in mathematics). p6 "subtracting all elements of one set from another" sounds confusing to me, as if you are calculating xy for each x in the first set and y in the second. I would consider "removing" p7 The group axioms \(e\circ g=g\) needs to be \(g\circ e=g\), or state it and the next one both ways around. p7 under multiplication the units of a commutative ring with 1 form an abelian group, but these are not all the elements of the ring (except in the trivial case where the ring has just 1 element). p8 If you want to be able to relax condition 11 then you need to write condition 12 both ways around! p8 A field is also required to have 1 not equal to 0. I'm out of time now  I'll look again in the week. 

20201012, 05:33  #14  
Sep 2002
Database er0rr
2·1,723 Posts 
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I mention it now in the both the paragraph on integers and when arguing about the irrationality of root 2. Quote:
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Edit I have uploaded the latest copy  12 Oct. Last fiddled with by paulunderwood on 20201012 at 14:42 

20201015, 09:09  #15 
Dec 2012
The Netherlands
2×727 Posts 
Some feedback on the version dated 12 October 2020:
2.1 Primes. I find the first sentence confusing (though that may just be me). Your explanation of what prime numbers are at the start of chapter 1 was clearer. 2.2 Euclid's algorithm If you are going to use both "greatest common divisor" and "highest common factor", perhaps saying that they are 2 names for the same thing would make it clearer. You say the gcd is the last nonzero remainder but don't explain what to do if the 1st remainder was 0 already. You explain why the last nonzero remainder divides a and b but not why it must be the highest number to do so. 3. Pythagoras You show that a+b and ab are both even and that no prime except 2 divides them both, and you also have that their product is a square. You conclude that a+b and ab are each 2 times a square. But the prime factorizations of a+b and ab could each contain an even number of 2's  you need to rule that out as well before writing a+b=2s² and ab=2t². (You are of course making it harder for yourself by working in the integers instead of the Gaussian integers here. As you introduced complex numbers in chapter 1, you could consider using Gaussian integers.) 
20201015, 15:48  #16 
Sep 2002
Database er0rr
2·1,723 Posts 
I have tried to correct and clarify the text and arguments as Nick pointed out in the previous post. I have uploaded the latest version  Oct 15th, 2020  to the original post.
I have not written about Gaussian integers, trying to restrict the text to natural primes, 
20201017, 10:00  #17 
Dec 2012
The Netherlands
10110101110_{2} Posts 
Feedback on version dated 15 October 2020:
7 Mersenne Numbers Formula for factorization of \(M_{pq}\) is almost right! 9.1 Legendre symbol Say p is an odd prime or p not equal to 2. Typo bottom of page 37: elements of A should go up to 10 instead of 11. 12 Frobenius Example: taking the polynomial \(x^2+1\), you get the Gaussian integers modulo n after all! Obviously it's up to you, but it might be worth including something on the Chinese Remainder Theorem. It would make it easier to explain your formula for the Euler phi function. Also, as you have a nice emphasis on the computational side of things in this, you could show how it is used in practice to speed up RSA decryption, for example. Just a thought, anyway. 
20201017, 23:57  #18  
Sep 2002
Database er0rr
D76_{16} Posts 
Quote:
CRT: I find that quite difficult. RSA: I have written that s or t may be chosen to be small for quick encoding or quick decoding respectively. Thanks again for your input. 18th Oct, 2020 version uploaded Last fiddled with by paulunderwood on 20201017 at 23:59 

20201018, 00:35  #19 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·643 Posts 
I love the book Paul.
Thank you very much. 
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