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Results tagged with nt.number-theory
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user 1587
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
34
votes
Does anyone know a polynomial whose lack of roots can't be proved?
Something close to what you want is in the paper
"Universal Diophantine Equation" by James P. Jones in the
Journal of Symbolic Logic 47 (1982), pp. 549--571.
Jones produces an explicit list of 37 eq …
- 12.4k
9
votes
Meaning of Kronecker's comment to Lindemann
I don't have a good answer to this question, in fact I don't think there is a good answer, because Kronecker did not always oppose the use of irrational numbers. For example, in his 1863 paper "Über d …
- 12.4k
17
votes
What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of...
Franz Lemmermeyer is better qualified than I am to answer
this question, and he certainly knows all about Kummer's motivation from
reciprocity laws. However, to answer some parts of your question: Ku …
- 12.4k
10
votes
What is the high-concept explanation on why real numbers are useful in number theory?
A possible candidate for a "minimal" result about integers that is a "projection" of a
result about reals: the group structure of the solutions of the Pell equation
$x^2-dy^2=1$ for $d$ a nonsquare po …
16
votes
Why are topological ideas so important in arithmetic?
May I suggest that we don't have to consider cohomology to see the influence
of topology on arithmetic? Looking for rational points on curves leads to the
question of which curves have rational parame …
- 12.4k
19
votes
8
answers
6k
views
To what extent is it true that "number theory = mathematics"? [closed]
In a thought-provoking answer to this MO question, Kevin Buzzard
and several commentators have described a multitude of ways in which
number theory is related to other parts of mathematics. It seems t …
15
votes
Fermat numbers and the infinitude of primes
It's interesting that the coprimality of Fermat numbers was already known in
Goldbach's time. The reason for attributing the proof to Polya is presumably
that such a proof is indicated as an exercise …
- 12.4k
20
votes
Knuth's intuition that Goldbach might be unprovable
There are also some concrete examples in graph theory, such as Kruskal's tree
theorem and the Robertson-Seymour graph minor theorem. These theorems
about infinite sequences of graphs were actually pro …
- 12.4k
54
votes
Accepted
New proofs to major theorems leading to new insights and results?
Here are a few examples from the 19th century.
Unsolvability of the quintic equation. Abel (1826) proved this by algebraic
ingenuity, but without clarifying the concepts involved. Galois (1830) gave …