Open-problem statements, with their sources attached.

Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers.

Dickson's conjecture* If a finite set of linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition, there exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all $i$.

Polignac's conjecture* For any integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.

The infinitude of Sophie Germain primes* There are infinitely many primes $p$ such that $2p + 1$ is prime.

The infinitude of cousin primes* There are infinitely many primes $p$ such that $p + 4$ is prime.

The infinitude of sexy primes* There are infinitely many primes $p$ such that $p + 6$ is prime.

The Elliott–Halberstam conjecture: for every $\theta < 1$ and $A > 0$ there exists a constant $C > 0$ such that $\sum_{1 \le q \le x^{\theta}} E(x; q) \le \frac{C x}{\log^A x}$ for all $x > 2$.

Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.

From [PPVW2016], Section 3.1: "from the mid-1960s to the present, it seems that most experts conjectured unboundedness."

From [PPVW2016], Section 8.2: "Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21". In other words, there are only finitely many elliptic curves over ℚ (up to isomorphism) with rank greater than 21. Notice that this contradicts the previous conjecture.

[PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank r and naïve height at most H is asymptotically H ^ ((21 - r) / 24 + o(1)). Note: ℰ_H in 8.2(b) should be ℰ_{≤H}, see the statement of Theorem 7.3.3. When r = 1, the exponent is 20 / 24 = 5 / 6, which agrees with the exponent in card_heightLE_div_pow_five_div_six_tensto and is consistent with half_rank_zero_and_half_rank_one.

[PPVW2016] 8.2(c): the number of elliptic curves over ℚ with rank ≥ 21 and naïve height at most H is asymptotically at most H ^ o(1).

The rank of the Elkies-Klagsbrun curve is exactly 29.

The rank of the Elkies curve is exactly 28.

It is not known whether there is an inifinite number of prime Euclid numbers.

It is not known whether every Euclid number is a square-free number.

Is there a perfect Euler brick?

Is there an Euler brick in $4$-dimensional space?

Is there an Euler brick in $n$-dimensional space for any $n > 3$?

Euler's sum of powers conjecture states that for integers $n > 1$ and $k > 1$, if the sum of $n$ positive integers each raised to the $k$-th power equals another integer raised to the $k$-th power, then $n ≥ k$.

The conjecture is known to be false for $k = 4$ and $k = 5$, but remains open for $k ≥ 6$.