<section xml:id="open-problems" label="section-open-problems">
<title>Open Problems</title>
<p>
Like for mathematical research.
An <c>openproblem</c>, <c>openquestion</c>, or <c>openconjecture</c> is numbered as a block, sharing the overall block counter by default, though a publisher can give them a counter of their own.
</p>
<p>
Beyond a <c>statement</c>, an open problem can be followed by discussion appendages such as <c>status</c>, <c>discussion</c>, <c>opinion</c>, <c>suggestion</c>, or <c>context</c>, and, like a <c>project</c>, it may be divided into <c>task</c>s.
</p>
<openproblem>
<statement>
<p>
Solve the Riemann Hypothesis<fn>Footnotes were once incomplete on open problems.</fn> <em>and</em> provide a short proof of Fermat's Last Theorem.
</p>
</statement>
</openproblem>
<openquestion>
<statement>
<p>
Is every even integer greater than two the sum of two primes?
</p>
</statement>
<status>
<p>
Verified by Oliveira e Silva for every even integer below <m>4\times10^{18}</m>.
The ternary version<mdash/>every odd number above five is a sum of three primes<mdash/>was settled by Helfgott in 2013, but the binary question is the one our seminar is after.
</p>
</status> <discussion>
<p>
The circle method predicts on the order of <m>n/(\log n)^2</m> representations of a large even <m>n</m>, comfortably positive; the stubborn obstruction is controlling the minor-arc contribution, where we have made no real progress.
</p>
</discussion>
</openquestion>
<openconjecture>
<title>Sum-free Sets of Squares</title>
<introduction>
<p>
This is the current focus of a project with our collaborators.
Write <m>S_N = \{1, 4, 9, \ldots, N^2\}</m> for the first <m>N</m> perfect squares, and call a set <term>sum-free</term> when it contains no solution of <m>a + b = c</m>.
We conjecture the following, listed in increasing order of difficulty.
</p>
</introduction>
<task>
<statement>
<p>
There is an absolute constant <m>c \gt 0</m> and, for every large <m>N</m>, a sum-free subset <m>A \subseteq S_N</m> with <m>|A| \geq (1/2 + c)\,N</m>.
</p>
</statement>
<status>
<p>
For every <m>N \leq 60</m> the largest sum-free subset we have found has exactly the size of the odd squares, so we cannot yet exhibit a single configuration that provably beats that baseline.
</p>
</status> <discussion>
<p>
The odd squares <m>\{1, 9, 25, \ldots\}</m> are already sum-free, since a sum of two of them is congruent to <m>2</m> modulo <m>4</m> while no square is; this gives density <m>1/2</m>, so the whole content of the conjecture is the gain <m>c</m>.
</p>
</discussion>
</task>
<task>
<statement>
<p>
The limiting density <m>d = \lim_{N \to \infty} \frac{1}{N} \max_{A} |A|</m> exists.
</p>
</statement>
<suggestion>
<p>
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
</p>
</suggestion>
</task>
<task>
<statement>
<p>
Replacing the squares by the cubes <m>\{1, 8, 27, \ldots\}</m> yields the same limiting density <m>d</m>.
</p>
</statement>
</task>
<conclusion>
<p>
Even the bare existence claim of the second part is open; we would be glad of either a proof or a numerical counterexample.
</p>
</conclusion>
</openconjecture>
</section>
Section 28 Open Problems
View Source for section
Like for mathematical research. An
openproblem, openquestion, or openconjecture is numbered as a block, sharing the overall block counter by default, though a publisher can give them a counter of their own.
Beyond a
statement, an open problem can be followed by discussion appendages such as status, discussion, opinion, suggestion, or context, and, like a project, it may be divided into tasks.
Open Problem 28.1.
View Source for openproblem
<openproblem>
<statement>
<p>
Solve the Riemann Hypothesis<fn>Footnotes were once incomplete on open problems.</fn> <em>and</em> provide a short proof of Fermat's Last Theorem.
</p>
</statement>
</openproblem>
Solve the Riemann Hypothesis and provide a short proof of Fermat’s Last Theorem.
1
Footnotes were once incomplete on open problems.
Open Question 28.2.
View Source for openquestion
<openquestion>
<statement>
<p>
Is every even integer greater than two the sum of two primes?
</p>
</statement>
<status>
<p>
Verified by Oliveira e Silva for every even integer below <m>4\times10^{18}</m>.
The ternary version<mdash/>every odd number above five is a sum of three primes<mdash/>was settled by Helfgott in 2013, but the binary question is the one our seminar is after.
</p>
</status> <discussion>
<p>
The circle method predicts on the order of <m>n/(\log n)^2</m> representations of a large even <m>n</m>, comfortably positive; the stubborn obstruction is controlling the minor-arc contribution, where we have made no real progress.
</p>
</discussion>
</openquestion>
Is every even integer greater than two the sum of two primes?
Status 28.2.1.
View Source for status
<status>
<p>
Verified by Oliveira e Silva for every even integer below <m>4\times10^{18}</m>.
The ternary version<mdash/>every odd number above five is a sum of three primes<mdash/>was settled by Helfgott in 2013, but the binary question is the one our seminar is after.
</p>
</status>
Verified by Oliveira e Silva for every even integer below \(4\times10^{18}\text{.}\) The ternary version—every odd number above five is a sum of three primes—was settled by Helfgott in 2013, but the binary question is the one our seminar is after.
Discussion 28.2.2.
View Source for discussion
<discussion>
<p>
The circle method predicts on the order of <m>n/(\log n)^2</m> representations of a large even <m>n</m>, comfortably positive; the stubborn obstruction is controlling the minor-arc contribution, where we have made no real progress.
</p>
</discussion>
The circle method predicts on the order of \(n/(\log n)^2\) representations of a large even \(n\text{,}\) comfortably positive; the stubborn obstruction is controlling the minor-arc contribution, where we have made no real progress.
Open Conjecture 28.3. Sum-free Sets of Squares
View Source for openconjecture
<openconjecture>
<title>Sum-free Sets of Squares</title>
<introduction>
<p>
This is the current focus of a project with our collaborators.
Write <m>S_N = \{1, 4, 9, \ldots, N^2\}</m> for the first <m>N</m> perfect squares, and call a set <term>sum-free</term> when it contains no solution of <m>a + b = c</m>.
We conjecture the following, listed in increasing order of difficulty.
</p>
</introduction>
<task>
<statement>
<p>
There is an absolute constant <m>c \gt 0</m> and, for every large <m>N</m>, a sum-free subset <m>A \subseteq S_N</m> with <m>|A| \geq (1/2 + c)\,N</m>.
</p>
</statement>
<status>
<p>
For every <m>N \leq 60</m> the largest sum-free subset we have found has exactly the size of the odd squares, so we cannot yet exhibit a single configuration that provably beats that baseline.
</p>
</status> <discussion>
<p>
The odd squares <m>\{1, 9, 25, \ldots\}</m> are already sum-free, since a sum of two of them is congruent to <m>2</m> modulo <m>4</m> while no square is; this gives density <m>1/2</m>, so the whole content of the conjecture is the gain <m>c</m>.
</p>
</discussion>
</task>
<task>
<statement>
<p>
The limiting density <m>d = \lim_{N \to \infty} \frac{1}{N} \max_{A} |A|</m> exists.
</p>
</statement>
<suggestion>
<p>
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
</p>
</suggestion>
</task>
<task>
<statement>
<p>
Replacing the squares by the cubes <m>\{1, 8, 27, \ldots\}</m> yields the same limiting density <m>d</m>.
</p>
</statement>
</task>
<conclusion>
<p>
Even the bare existence claim of the second part is open; we would be glad of either a proof or a numerical counterexample.
</p>
</conclusion>
</openconjecture>
This is the current focus of a project with our collaborators. Write \(S_N = \{1, 4, 9, \ldots, N^2\}\) for the first \(N\) perfect squares, and call a set sum-free when it contains no solution of \(a + b = c\text{.}\) We conjecture the following, listed in increasing order of difficulty.
(a)
View Source for task
<task>
<statement>
<p>
There is an absolute constant <m>c \gt 0</m> and, for every large <m>N</m>, a sum-free subset <m>A \subseteq S_N</m> with <m>|A| \geq (1/2 + c)\,N</m>.
</p>
</statement>
<status>
<p>
For every <m>N \leq 60</m> the largest sum-free subset we have found has exactly the size of the odd squares, so we cannot yet exhibit a single configuration that provably beats that baseline.
</p>
</status> <discussion>
<p>
The odd squares <m>\{1, 9, 25, \ldots\}</m> are already sum-free, since a sum of two of them is congruent to <m>2</m> modulo <m>4</m> while no square is; this gives density <m>1/2</m>, so the whole content of the conjecture is the gain <m>c</m>.
</p>
</discussion>
</task>
There is an absolute constant \(c \gt 0\) and, for every large \(N\text{,}\) a sum-free subset \(A \subseteq S_N\) with \(|A| \geq (1/2 + c)\,N\text{.}\)
Status 28.3.a.1.
View Source for status
<status>
<p>
For every <m>N \leq 60</m> the largest sum-free subset we have found has exactly the size of the odd squares, so we cannot yet exhibit a single configuration that provably beats that baseline.
</p>
</status>
For every \(N \leq 60\) the largest sum-free subset we have found has exactly the size of the odd squares, so we cannot yet exhibit a single configuration that provably beats that baseline.
Discussion 28.3.a.2.
View Source for discussion
<discussion>
<p>
The odd squares <m>\{1, 9, 25, \ldots\}</m> are already sum-free, since a sum of two of them is congruent to <m>2</m> modulo <m>4</m> while no square is; this gives density <m>1/2</m>, so the whole content of the conjecture is the gain <m>c</m>.
</p>
</discussion>
The odd squares \(\{1, 9, 25, \ldots\}\) are already sum-free, since a sum of two of them is congruent to \(2\) modulo \(4\) while no square is; this gives density \(1/2\text{,}\) so the whole content of the conjecture is the gain \(c\text{.}\)
(b)
View Source for task
<task>
<statement>
<p>
The limiting density <m>d = \lim_{N \to \infty} \frac{1}{N} \max_{A} |A|</m> exists.
</p>
</statement>
<suggestion>
<p>
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
</p>
</suggestion>
</task>
The limiting density \(d = \lim_{N \to \infty} \frac{1}{N} \max_{A} |A|\) exists.
Suggestion 28.3.b.1.
View Source for suggestion
<suggestion>
<p>
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
</p>
</suggestion>
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
(c)
View Source for task
<task>
<statement>
<p>
Replacing the squares by the cubes <m>\{1, 8, 27, \ldots\}</m> yields the same limiting density <m>d</m>.
</p>
</statement>
</task>
Replacing the squares by the cubes \(\{1, 8, 27, \ldots\}\) yields the same limiting density \(d\text{.}\)
Even the bare existence claim of the second part is open; we would be glad of either a proof or a numerical counterexample.
