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5 of 31 people found the following review helpful
2.0 out of 5 stars There is no such thing as 'number theory'!, 8 Aug 2010
This review is from: An Introduction to the Theory of Numbers (Paperback)
I'm not as impressed as the other reviewers here with this book, despite it's being in some sense a 'classic'.

I take it some people - I have no idea what proportion of the population - now and then are struck by properties of numbers. For example, if a number AB is added to BA, the result always divides by 11. If the difference is work out, this is always a multiple of 9. Or (e.g.) Any three consecutive numbers, say 6,7, and 8, if multiplied together, give a result which divide by 6. This sort of thing is the basis of 'number theory'.

There are at least two problems with this book. Firstly, there is in fact not yet such a thing as 'number theory'. This book is a ragbag of techniques and things which have been identified and passed down by lecturers. But it is NOT a coherent 'theory' in any sense. Perhaps I might compare it with a book on 'chess theory'. Chess books have accounts of such things as opening gambits, sacrifices, end games - including some with extremely precise techniques needed for victory. And there are things like 'zwischenzug' and assorted events which are rare, but have some interest. But does this make up a body of 'theory'? I'd say not. Anyone looking to this book for insight into the Pythagorean mystery of number will in my view be more or less disappointed.

Now, what follows from this is my second point, which perhaps is to do with human psychology, or the capacity of the human brain. What is it that makes some people fix on a certain type of problem? For example, this book, like most or maybe all on number theory, starts with prime numbers - probably discovered as a result of packaging and division of actual objects. This of course had practical applications, such as the Babylonian 360 degrees, and our 12 inches, 14 pounds, 1760 yards, and so on. A collection of techniques (e.g. Eratosthenes' sieve), formulas, limits and other results has accumulated. Looking at Euclid's proof of the infinity of primes, his method was to multiply all the primes, and add 1. This function in effect is designed to use the properties of primes to generate a new prime. However Hardy and Wright don't attempt to generalise this process. Maybe Fermat's Last Theorem could be proved elegantly by inventing some ingenious function which combines the properties of addition, multiplication, and powers - repeated multiplication by the same number? What is it that makes some problems (so far) insoluble - and many of them are very trivial to state?

So we have here a collection of results, embodied in symbolism which is far enough from the actualities to (perhaps) look more impressive than it really is. Integration, for example, is basically simple enough, but the long s and the notation removes the reader from the real world...

And there's a related problem, which is that the connective material, explaining why the next bit is there and what it is supposed to illustrate, is completely missing. The result is like a tour of museum exhibits, where the tourist is expected to infer the significance of all the specimens. Or like a concert, where one sample piece of music is played after another, from which the auditor is presumably left to infer a theory of music. In fact, I've just decided to demote the book to two stars!
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Tracked by 2 customers

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Showing 1-10 of 15 posts in this discussion
Initial post: 9 Aug 2010 11:34:55 BDT
Sorry, this wasn't edited properly - there's a better version on amazon dot com! I don't want to change it in case the whole thing gets lost

Posted on 26 Aug 2010 12:14:25 BDT
This was on the dot com site, Aug. 12, 2010 10:48 AM PDT, Shay Allen Logan of Spokane says:

This review is absolutely fascinating. Garbage, and poorly informed, but extremely fascinating. I would like it if the author could explain further what he/she means by "the Pythagorean mystery of number". Also, what does the author have in mind with the "ingenious function"? A moment's study of abstract algebra while show we have, in fact, studied things in much more generality than it would appear you think we have, though I could be misunderstanding you.

Lastly, what is it about integration that is so simple? If you could explain to me what is meant by this comment, I would possibly find your review far more credible, and less like a rambling tirade from an ignorant person unhappy with something they have no understanding of.

-------Mr (Ms?) Logan -- the usual proof of an infinite number of primes is based on a function. This function is all the primes, up to and including the possible biggest, being multiplied together. Then 1 is added. This is ingenious - it uses the properties of composite numbers, so we can be certain the product (without 1 added) is divisible by every single prime up to the possible biggest. Then when 1 is added we can be certain not one of these primes now divides into it. Therefore, the new number is either a prime or has factors each bigger than the supposed biggest prime.

I wondered if Fermat's proof of his last theorem - if he had one - used some similar ingenious function which combined powers and addition in some way to clearly rule out any solution greater than two.

-------
Integration just means adding up, as the long s is intended to suggest. In Newton's time, the letter s at the start of a word was written large - a bit like an f. Hence the integration symbol, still in use today.

Posted on 11 Oct 2011 11:52:57 BDT
Shlee says:
This reviewer has obviously never studied maths. This book outlines the core theories in divisibility, prime numbers, congruences, modulo arithmetic, arithmetical functions and their behaviour, i.e. what we have been learning since we first started school! The fact that he clearly has not understood it speaks a lot about his intellect on this subject. The 'long s' that he complains about is widely used notation for integration, as most people reading this book will know, and not just something that Hardy and Wright made up. And his attempted generalisation of Euclid's Theorem shows that he has not fully understood that this proof proves there are infinitely many primes, by contradiction of assumption, and not just a way to 'generate a new prime'. Perhaps he should learn the meaning of a mathematical theory, before waving away the fact that the study of number theory has existed for 5-6000 years? I invite him to learn the basics of mathematics, before picking up a book wildly beyond his abilities and scrutinising it.

In reply to an earlier post on 14 Oct 2011 18:23:19 BDT
'Shlee' says:
____________________________________
This reviewer has obviously never studied maths. This book outlines the core theories in divisibility, prime numbers, congruences, modulo arithmetic, arithmetical functions and their behaviour, i.e. what we have been learning since we first started school! The fact that he clearly has not understood it speaks a lot about his intellect on this subject. The 'long s' that he complains about is widely used notation for integration, as most people reading this book will know, and not just something that Hardy and Wright made up. And his attempted generalisation of Euclid's Theorem shows that he has not fully understood that this proof proves there are infinitely many primes, by contradiction of assumption, and not just a way to 'generate a new prime'. Perhaps he should learn the meaning of a mathematical theory, before waving away the fact that the study of number theory has existed for 5-6000 years? I invite him to learn the basics of mathematics, before picking up a book wildly beyond his abilities and scrutinising it.
________________________________________

I'm quite amused by this rubbish. I have a maths degree, and, unlike you, an interest in how maths might develop in future. This review was written for people who may perhaps have picked up a volume on maths, and wondered what it's about. Perhaps you should stop preening yourself on being able to parrot what you've been told, and try to consider the subject anew.

In reply to an earlier post on 13 Nov 2011 18:51:07 GMT
A. P. Thomas says:
I would have to agree with Shlee; though I have no reason to doubt Rerevisionist has a maths degree and an interest in the subject, he comes across as somewhat bitter in his review.
---------------------------------
"Firstly, there is in fact not yet such a thing as 'number theory' "
---------------------------------
If I understand what you are trying to say then I agree with your meaning, though it is entirely irrelevant! Number Theory IS a collection of ragtag theorem and conjecture. There is not and will never be ONE theory of numbers and this in no way diminishes what is already a beautiful subject to those who research and enjoy it.

Number theory is pretty much dependent on prime numbers due to the fact they are in essence the building blocks of all other (compound) numbers. This is why they are so fascinating. Number theory was never intended to have a purpose or an application; this was something that, as a pacifist, G.H. Hardy was quite proud of. Though of course as things have developed many uses have been made of number theories, with a famous example being the Enigma / Tunney code breaking during the 2nd World War.

I'm not sure if you are aware, but Fermat's Last Theorem has been proven by Andrew Wiles before the time of you writing your review. The proof is around 42 pages long and uses maths only recently developed. There is very little chance of it being solved by some ingenious yet straightforward application of the basic functions of arithmetic.

This is one of the fascinating aspects to number theory and indeed other parts of mathematics. That a problem that can be stated so simply and can, on the face of it, be understood by so many, can prove almost impossible to even the greatest minds that have tackled them.

On one last note, this book is actually titled as "an INTRODUCTION to the theory of numbers". It serves this purpose extremely well; your dislike of some of the language in the book strikes me more as a personal preference than a problem with the content. The Pythagorean mystery of number is merely a reference to the classical period of the famous Greek mathematicians.

Posted on 19 Apr 2012 07:19:39 BDT
Last edited by the author on 19 Apr 2012 07:21:11 BDT
[Customers don't think this post adds to the discussion. Show post anyway. Show all unhelpful posts.]

In reply to an earlier post on 19 Apr 2012 13:05:12 BDT
Last edited by the author on 19 Apr 2012 13:07:53 BDT
Hi J McDonnell junior. Perhaps you could explain how you can be sure there can be no overarching theory of numbers? Possibly the subject, such as it is, is in a condition analogous to chemistry before the periodic table - i.e. a lot of observations and deductions, and suggestive comparisons, but without something to hold them together.

I don't know whether to wish you luck in trying to nestle into a career of quoting other people, or wish you'd try something creative. There's something to be said both ways. All the other contributors seem to be in the first category.

The three consecutive numbers thing wasn't very well worded - and you didn't see what I was getting at. The point here is that a cube, minus the original number, is always a multiple of 6. (Formally, ish: (n-1), n, (n+1) are consecutive, and multiply to n^3 - n. And at least one must be a multiple of 3. Etc). All the results of number theory must, presumably, have been found by people trying things out or stumbling across them.

In reply to an earlier post on 19 Apr 2012 17:36:41 BDT
I think that you should define EXACTLY what you mean by "overarching" before I can answer your first question. Also,
I do understand why n^3-n is divisible by 6, but I do not feel that my response fails to grasp this from your review, as this part is literally taken from your review, and the subsequent interpretation is derived from it.
What was your degree in?

In reply to an earlier post on 21 Apr 2012 18:21:36 BDT
Hey - I'm being polite. But I'm not interested. You're not getting the point.

In reply to an earlier post on 23 Apr 2012 18:24:10 BDT
Last edited by the author on 23 Apr 2012 18:24:34 BDT
I get the point, I simply don't agree with it, and tried to convince you otherwise. Now I realise further argument is futile. Thank you.
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