50 problems

A more serious problem states as follows: Schoenberg proved about 70 years ago that $\varphi(n)$ has a distribution function. In other words the density of the integers for which…

Let $a_1, \ldots, a_t$ be the longest sequence for which … Probably $t = \pi(n)$. Can one even prove $t < (1 + o(1))\pi(n)$ or at least $t = o(n)$? This latest conjecture will prob…

Euler's $\varphi(n)$ function is the number of integers $1 \leq t < n$ relatively prime to $n$, and $\sigma(n)$ is the sum of divisors of $n$. Is it true that for infinitely many i…

Perhaps this is really a Turán type problem and not a Ramsey problem. In other words, if $m$ is sufficiently large and $1 < a_1 < a_2 < \cdots < a_\ell \leq m$ is a sequence of int…

An old problem of R.L. Graham and myself states: Is it true that if $m_k$ is sufficiently large and we colour the integers $2 \leq t \leq m_k$ by $k$ colours then … is always solva…

One last Ramsey type problem: Let $n_k$ be the smallest integer (if it exists) for which if we colour the proper divisors of $n_k$ by $k$ colours then $n_k$ will be a monochromatic…

An old problem of mine states as follows: Let $a_1 < a_2 < \cdots < a_k$ be a Sidon sequence. Can one extend it to a larger Sidon sequence … In other words, loosely speaking: can o…

More generally: Let $a_1 < a_2 < \cdots < a_{k_1} \leq n$, $b_1 < b_2 < \cdots < b_{k_2} \leq n$ be two Sidon sequences for which $a_j - a_i \neq b_v - b_u$ for all $i < j$ and…

Let $1 \leq a_1 < a_2 < \cdots < a_h \leq n$ be a maximum Sidon sequence. Can one find a Sidon sequence $b_1 < b_2 < \cdots < b_r \leq n$ for every $r$ and $n > n_0(r)$ so that the…

Here is an old conjecture of mine: Let $a_1 < a_2 < \ldots$ be an infinite sequence for which all the triple sums $a_i + a_j + a_k$ are distinct. Is it then true that…

A sharpening of our old conjecture with Turán would state: If $a_n < Cn^2$ for all $n$ then $\limsup f(n) = \infty$. In fact, for what functions $g(n) \to \infty$ does…

Sidon also asked: Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence for which all the sums $a_i + a_j$ are distinct. Put … Probably there is a Sidon sequence $A$ for which ……

It would also be of interest to decide whether there exists a sequence $b_1 < b_2 < \ldots$ which is not a basis and which has the following property: If $a_1 < a_2 < \ldots$ is a…

A sequence A is called by Khintchin an essential component if for every sequence B with $d_s(\mathrm{B}) > 0$, $d_s(\mathrm{A} + \mathrm{B}) > d_s(\mathrm{B})$. [...] This led me t…

A somewhat similar question is the following one: Let $a_1, a_2, \ldots, a_{2n}$ be $2n$ integers in the interval $(1, 4n)$ and let $b_1, b_2, \ldots, b_{2n}$ be the other $2n$ int…

The sequence B is called a basis of order $k$, if every integer is the sum of $k$ or fewer $b$-s. Let $d_s(\mathrm{A}) = \alpha$ then I proved that … Thus every base is an essentia…

Let $a_1 < \ldots < a_k \leqslant n$ be a sequence of integers and form all sums $\sum_{i=u}^{v} a_i$. Can one have $cn^2$ distinct numbers in this set for some $c > 0$? This does…

Let $b_1 < b_2 < \ldots$ be an infinite sequence of integers, so that every integer is of the form $b + k^2$. Denote by $B(x)$ the number $b$-s not exceeding $x$. [...] I can not d…

Let $a_1 < a_2 < \cdots$ be an infinite sequence of integers. Straus and I conjectured that there is a sequence of density $0$, $b_1 < \cdots$ so that every integer is of the form…

Let $a_1 < a_2 < \ldots$ be any infinite sequence of integers. Straus and I conjectured that there always exists a sequence $b_1 < b_2 < \ldots$ of density $0$, so that every suffi…

Sidon also asked: Let $1 \leq a_1 < a_2 < \cdots < a_k \leq n$ and assume that $a_i + a_j$ are all distinct. Put $h(n) = \max k$. Determine or estimate $h(n)$ as accurately as poss…

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also I offer 100 dollars for an explicit cons…

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also Turán and I conjectured that if…

Let us now restrict ourselves to a special case. The $n_i$ are the integers between $t$ and $ct$. First of all denote by $\alpha_1(c,t)$ the smallest possible value of the density…

Tenenbaum and I recently asked the following question: let $n_1 < n_2 < \ldots$ be an infinite sequence of positive integers. Is it then true that there always is a positive intege…