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A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective …

The answer is "yes" - the order mod p of 2 is almost always as large as the square root of p (actually you get epsilon less than this in the exponent). If you take r multiplicatively independent numbe …

I just don't think it's true, despite my own tastes in topics. Such formulations are substantially a matter of fashion. There is one basic axis, running from very detailed information at one end (wh …

This is a classical Diophantine equation (Mordell, Diophantine Equations, p. 258). Apart from n = 0, 1, -1, there is only the solution n = 24. Proofs by G. N. Watson (1919), W. Ljunggren (1952). Ther …

Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ri …

The motivation "because I'd like to know [...] if it is impossible to prove FLT using elementary methods" seems to require comment. It is much more likely (in my view) that it is true that FLT can be …

I think, tautologously, any method proving the Riemann Hypothesis (or even seriously improving our knowledge on the zeroes) becomes "number theory" immediately. That said, I know what the question mea …

Bertrand's postulate (http://en.wikipedia.org/wiki/Bertrand%27s_postulate).

Bryan Birch's view is that they form a bottomless area for research problems. All answers to the question fall into two types: showing examples of why this is true, and asking why it should be true. G …

You have a typical recursively enumerable set S of integers, and a set X of lattice points cut out by a multivariate polynomial. We are talking about S being the projection (onto one axis) of X. Given …

One way of looking at the issue is this: it is quite easy to transform the question of good rational approximations to algebraic numbers into a question about integral points on certain affine curves …

The thing is that it is well known that for the quadratic Gauss sums, expressed as an exponential sum rather than with Legendre symbols, the path is very much not a random walk when you plot it in the …

I think your statement could usefully be sharpened in a couple of ways. Firstly, the state-of-the-art is that true statements for function fields are expected to have analogues for number fields. Th …

I have had this question on my mind for two decades. We know, after Heath-Brown, that one out (say) of 3, 5, 7 is a primitive root mod p for infinitely many primes p. We just don't know which one. (We …

From the point of view of analytic number theory, the point usually would be asymptotic behaviour. This is typically well understood, and is in the massive book of Watson. Apart from that, yes, numero …