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Results tagged with nt.number-theory
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user 99045
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Coin flipping game
To complete the proof of Dieter Kadlka:
For $n=2^m$, we are looking for $a\neq b< n$ such that $$ b^2+b=a^2+a+l 2^{m+1} $$ for some $l\in \mathbb{N}$ and then $$l 2^{m+1} = (b-a)(b+a+1)$$
Because $b-a …
- 4,361
18
votes
Sets of unit fractions with sum $\leq 1$
The bound proposed by Lucia is correct. I add some detail and I stress that It is an application of "large deviation" theorem which is very very standard.
Set the following independent Bernoulli rand …
- 4,361
4
votes
Accepted
How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?
$$\mathbb{E}[\text{#|roots of P|}]=\mathbb{E}(\sum_{k\in \mathbb{Z}}1_{k \text{ is a root of P}})=\sum_{k\in\mathbb{Z}}\mathbb{P}(P(k)=0)$$
For $k=0$, $\mathbb{P}(P(0)=0)=\frac{1}{(2B+1)}$.
For $k\ne …
- 4,361