Recent content by A. Bahat

This isn't quite what he proved. Rather than a bound on the distance between twin prime pairs, the new result is that there are infinitely many pairs of primes differing by at most 70 million. If we can get 70 million down to 2, then that's a different story.

While I also think Rudin's exposition is pretty poor from a pedagogical perspective, many of the problems are quite good. It's a good book to have, but maybe not the best to learn from. Since you seem pretty comfortable with rigorous calculus, I think you wouldn't have trouble with Rudin. Just...

I'm not a huge fan of this book. While it has some nice explanations, the coordinates are overwhelming! Sums and bases and indices, oh my! E.g. determinants make up the first chapter, which strikes me as odd, and as such, the chapter ends up very computational--he hasn't even defined a linear...

Spivak really loves differential geometry, as these books show (I will restrict myself to the first two volumes, for I am unfamiliar with the rest). Everything is motivated with the utmost care--all the abstract topological stuff in the first volume is made completely natural in setting up the...

Hartshorne is kind of like a super-condensed version of Grothendieck's EGA. The actual geometry (not that schemes aren't geometry, but you know what I mean--the concrete stuff from which the motivation for the general theory derives) is in Chapters 1, 4, and 5, on Varieties, Curves, and...

I really like this book. There are relatively few examples and problems in Lang. However, when I think of this book, I think of the examples and problems as perhaps the best part! Here's why: although they are fewer in number, they are excellent. In the examples, you can find compact Riemann...

I have mixed feelings about this book. It can be quite useful as a reference and for some explanations; that I cannot deny. On the other hand, sometimes this book just seems like too much of an encyclopedia to me. It doesn't get me excited about algebra. I would think it ideal to start with...

This is where I first learned algebra. It is an excellent book, written in an "organic" style reminiscent of Arnold, Atiyah, Poincare, Riemann, etc.--the theory is always well-motivated, and abstraction for abstraction's sake is kept at bay. There are important topics not covered in Artin (dual...

Try Spivak's Calculus on Manifolds, or maybe Edwards's Advanced Calculus of Several Variables.

My two cents: if you already know trigonometry and vectors, you will do fine with calculus (assuming your algebra is up to par...); for an intro. mechanics textbook, something like Calculus Made Easy should suffice (I think). Halliday/Resnick is probably good, but go for an older edition if you...

An important part of learning e.g. advanced physics (the same applies to math, with which I have personal experience, and I imagine to many other fields as well) is getting confident in your solutions. Naturally, sometimes you will be unsure, especially at the beginning, but eventually you've...

I've got a copy of Shilov in front of me, and on page 2 while defining a field (or number field, as he calls it) he writes "The numbers 1, 1+1=2, 2+1=3, etc. are said to be natural; it is assumed that none of these numbers is zero." That is, he is only working with fields of characteristic...

I can't really comment on the feasibility of your plan, but I can address some of your concerns. Why should you need answers to get something out of the exercises? To be clear, I'm not saying, "Suck it up, answers are for wimps." But the fact that there are no answers doesn't make the...

Oh, and for algebra, I recommend Artin.

Rudin is pretty standard for introductory real analysis, but is probably too slick for optimal learning. Besides, it seems that Wade works in substantially less generality than Rudin (i.e. on the real line and later in Euclidean space rather than in metric spaces). Perhaps try Lang's...