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Results tagged with nt.number-theory
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user 405
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
Sequence with upper-bounded distance between same element
In computer science the first problem has been studied under the name of the "Pinwheel problem". A few observations/known facts:
A necessary condition is that $\sum (s_i+1)^{-1}$ is at most $1$ (si …
- 5,781
12
votes
Accepted
How many random sieve operations to decimate the set {2,...,n}?
An equivalent way of describing the process: We start with a randomly chosen permutation $\tau$ of $\{2, \dots, n\}$. At each step we choose the first number in $\tau$ which is still in our set $S$, …
- 5,781
18
votes
Accepted
Splitting Pythagorean triples
This problem appears in Croot and Lev's 2007 "Open Problems in Additive Combinatorics" (http://people.math.gatech.edu/~ecroot/E2S-01-11.pdf ), where it is attributed to Erdos and Graham (the latter of …
3
votes
Accepted
Average size of determinants of integer matrices?
As noted in Will's comment above, it's easy to compute the expected square of the determinant. More precisely, we have
$$E(\det M^2)=n! \prod \frac{k_i (k_i+1)}{3}.$$
Let $M'$ be formed from $M$ by …
- 5,781
2
votes
Accepted
Problem regarding subsets that sum to 0
As suggested by Christian, you may want to start by looking at the Littlewood-Offord problem. Here's a scaled version of Erdős' result that might be more relevant to your problem:
"If $a_1, \dots a_ …
- 5,781
10
votes
Accepted
Sets A such that A+A contains the largest set [0,1,..,t]
A table of values for these $t$ are given in the introduction Graham and Sloane's On Additive Bases and Harmonius Graphs (your sequence corresponds to $n_\beta(k)$ in their notation). Graham and Sloa …
- 5,781
4
votes
Accepted
Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold?
It seems the energy version is true if make the additional assumption that $\lambda=c/d$ is rational, meaning that $T_A(\lambda)$ counts the number of solutions in $A$ to
$$d(a_1-a_2)-c(a_3-a_4)=0 \, …
- 5,781
3
votes
Primes are pseudorandom?
One example of a conjectured statement of pseudo-randomness for the primes that formalizes what Tao describes in his slides is the Hardy-Littlewood conjecture for k-tuples, which roughly says that you …
- 5,781