This isn't a bad book, but don't let the title fool you. This is not a history of number theory. It is a textbook of number theory with a little bit of history tossed in. And a very large portion of that history is in the first chapter, and is devoted not to number theory as such but to the various systems used over the years to record numbers: Egyptian, Greek, Babylonian, and so forth.

Then comes the textbook. When you get to the actual mathematics, there is only the slightest history: "Euler discovered such-and-so," or "Fermat recorded," or "Gauss calculated...." And it left me with a serious question: Is this a number theory book, or an abstract algebra book? Admittedly the boundaries are so faint as to be non-existent, but the section on prime numbers (clearly number theory) is pretty thin, while the section on congruence arithmetic (which I, at least, would call abstract algebra) is so substantial that I think it exceeds what we did in my college abstract algebra class!

And surely construction problems are geometry, even if they're approached entirely algebraically. But even if they are considered algebraic... that's not arithmetic.

On the other hand, there is a good bit about Diophantine Equations, which probably qualify as number theory. So... I guess you can file the book wherever you like.

Just don't plan on reading it casually. It uses the standard advanced-textbook trick -- I think it's an attempt to prove how smart the author is, but it has never impressed me -- of not showing enough work to actually understand its proofs, forcing you to sweat hard to figure them out. (If you ever have to prove something for real, you will have to show your work! It's just a way to make students sweat without teaching them anything.) And the problems it assigns are fairly advanced, and rarely build (that is, you don't get easy problems that you can solve by inspection, letting you see the technique while you work your way up to the hard ones). I didn't really think it a particularly good set of exercises even if you set that aside. Nor are the really important results -- the ones you need to learn and remember -- marked out very clearly.

Plus, of course, it's three-quarters of a century old, and a lot of work has happened since then. Some of that is obvious (Fermat's Last Theorem has been proved), some less so -- e.g. all that congruence arithmetic is really important in computer contexts, but the uses here aren't all that relevant, and as for mentions of hexadecimal or octal math, and the effects of different number bases on how one performs calculations, forget about it.

I might rate this more highly if I hadn't expected something different -- from the title, I expected something with a lot more history, and a lot more actual manipulation of numbers, and frankly a lot more "beautiful" mathematics; I ordered it in hopes of having a little fun -- but even so, in the current era, I think this has some value as a supplement to an abstract algebra class, but it's really not fit to be the sole textbook of a number theory class. ( )