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### Non-asymptotically densest progression-free sets

For the context of this question, a progression-free set is a subset of integers that does not contain length-three arithmetic progressions.
For large $N$, it is known that $[N] = \{1, \ldots, N\}$ ...

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### Is there another proof for Dirichlet's theorem? [duplicate]

Possible Duplicate:
Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible?
Dirichlet's theorem on primes in arithmetic progression states that there ...

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### Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$).
Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$.
[The ...

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### Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies
$$
2n-1\leq |A+A|\leq \frac{n(n+1)}{2}.
$$
The lower bound is essentially ...

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### Sum of Series Where Exponent is Sum of Arithmetic Progression

Hi,
How do i get the sum of such a sequence:
$1 + x^{-1} + x^{-3} + x^{-6} + ...$
where the exponents are actually sum of arithmetic progression. i.e.
$x^{-0} + x^{-(0 + 1)} + x^{-(0 + 1 + 2)} + ...

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### Covering Systems of infinite sets of residue classes mod primes

Take an infinite set of distinct primes and a (edit: or 2 , etc.) residue class for every prime. For exammple you can take all the primes bigger than some prime or the primes of a specific form (i.e. ...

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### greatest common divisor of p-1 and q-1 [closed]

Hi there,
Can we say that if $p$ and $q$ are distinct prime number diving $n$
$\Omega(gcd(p-1,q-1)) \leq \Omega(n)$
Where $\Omega(n)$ denotes the number of prime powers dividing $n$
Best
rahmi

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### Arithmetic progression and most significant digits in different bases

Given a number $x \geq 3$, let $b(x) \in \{0,1\}$ be the second most significant digit (bit) of its binary representation, and $t(x)\in \{1,2\}$ the most significant digit of its ternary ...

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### sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by :
$$y_{n+1}=a y_n+b z_n+c$$
where $(z_n)$ is an arithmetico-geometric sequence :
$$z_{n+1}=d z_n+e$$
and $a,b,c,d,e$ real numbers.
Thank you ...

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### Do there exist sets of integers with arbitrarily large upper density which contains infinitely many elements that are not in an arithmetic progression of length 3?

Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of ...

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### Arithmetic progression and 3^m,3^{m+1} intervals

I'm trying to prove (or disprove) the following "conjecture".Given the following set of powers of two:
$$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$
...

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### Conjecture about distribution of primes in arithmetic progression

For my work, i need the following
Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, ...

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### Greedy sequences without k-term arithmetic progressions

If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...

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### Intersection of two arithmetic progressions

Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, where $n\in\mathbb{N}^0$, it can be shown that the intersection of these ...