Hall's conjecture: a square extremely close to a cube
Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.
问题
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符合的问题
Euler's sum of powers conjecture beyond fifth powers
Find k−1 positive k-th powers summing to a k-th power for some k ≥ 6, extending the counterexamples to Euler's conjecture for k = 4 and 5.
The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
The Erdős–Turán conjecture on additive bases
Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
The Erdős–Straus conjecture
Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
Elliptic curves with a prescribed number of points
Given a prime p > 2^80 and a target N with a large CM discriminant, quickly construct an elliptic curve over F_p with exactly N points.
Integers not overtaken by m + τ(m)
Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
Dense polynomials whose squares are sparse
Find a polynomial of degree n with all n+1 coefficients nonzero integers whose square has fewer than √n nonzero coefficients.
Dense infinite Sidon sets
Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.
The de Bruijn–Newman constant
Narrow the de Bruijn–Newman constant, which equals 0 exactly when the Riemann hypothesis holds; it is known that 0 ≤ Λ ≤ 0.2.
Composite numbers in sumsets of sparse sets
Show that if A, B ⊆ {1,…,N} each have at least N^0.49 elements, then the sumset A + B contains a composite number.
Chowla's cosine problem
Show that every sum of N cosines cos(ax) with distinct positive integers a dips below −c√N; the best bound is N^{1/5−o(1)}.
The Brocard–Ramanujan equation n! + 1 = m²
Prove that n! + 1 is a perfect square only for n = 4, 5 and 7, or find a fourth solution.
A composite number that passes the Baillie–PSW test
Find an odd composite number passing both the strong base-2 Fermat test and the strong Lucas test of Baillie–PSW; none is known below 2^64.