Open Problem 28.1.
Solve the Riemann Hypothesis and provide a short proof of Fermat’s Last Theorem.
1
Footnotes were once incomplete on open problems.
Section 28 Open Problems
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 Question 28.2.
Is every even integer greater than two the sum of two primes?
Status 28.2.1.
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.
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
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)
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.
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.
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)
The limiting density \(d = \lim_{N \to \infty} \frac{1}{N} \max_{A} |A|\) exists.
Suggestion 28.3.b.1.
One of our collaborators proposes a Fekete-style subadditivity argument to force the limit to exist, even before we can name its value.
(c)
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.
