Mergelyan's theorem with zero-free polynomials
Decide whether functions holomorphic and zero-free inside a compact set can be uniformly approximated by polynomials with no zeros on the set.
问题
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符合的问题
Wilf's conjecture on numerical semigroups
Find a numerical semigroup with e(S)·n(S) < c(S), refuting Wilf's 1978 conjecture, or prove it; it holds for all semigroups with up to 60 gaps.
The sum-product problem for integers
Show that every finite set of integers has nearly |A|² distinct sums or nearly |A|² distinct products; false over the reals since 2026.
Square-difference-free sets: the Furstenberg–Sárközy exponent
Find the growth exponent of the largest subset of {1,...,N} with no square difference, now known to lie between 0.7580758 and 1 - c.
The sparse ruler constant
Find the limit of F(N)/√N, where F(N) is the fewest marks on a ruler of length N that measures every distance; it lies in [1.56, √3].
Do six small cubic equations have infinitely many integer solutions?
Prove that each of six cubic equations z² + y²z + x³ + … = 0 of size at most 24, the smallest open cases, has infinitely many integer solutions.
Singmaster's conjecture on repeated binomial coefficients
Decide whether, for every t, some number appears as a binomial coefficient C(n,k) with k ≤ n/2 in exactly t ways.
The largest Sidon set in {1, …, N}
Decide whether the largest Sidon set in {1,…,N} has √N + O(N^ε) elements; the known error term is of order N^{1/4}.
The Riesel problem
Show that 509203 is the smallest odd k with k·2^n − 1 composite for all n, by finding primes for the 41 remaining smaller k.
Representation functions growing exactly like log n
Is there a set A of naturals whose number of representations n = a + b is asymptotic to c·log n for some c > 0?
Rado's boundedness conjecture
Show that a linear equation that some finite colouring of the integers avoids can be avoided with a number of colours depending only on its length.
Must x be an integer if 2^x and 3^x are integers?
Show that a real number x for which both 2^x and 3^x are integers must itself be an integer.
A polynomial bijection from Q × Q to Q
Decide whether some polynomial with rational coefficients maps Q × Q bijectively onto Q.
Primes p for which p − 2 has an odd number of prime factors
Prove that for infinitely many primes p, the number p − 2 has an odd number of prime factors counted with multiplicity.
Odd weird numbers
Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.
Covering systems with odd moduli
Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
The Erdős minimum overlap problem
Pin down the constant c ≈ 0.38 such that every split of {1,…,2N} into equal halves has a difference occurring cN times.
Arbitrarily long runs of equal signs of the Liouville function
Prove that the Liouville function takes the same sign on arbitrarily long blocks of consecutive integers.
The lonely runner conjecture
Find integer speeds for which no moment leaves every runner at least 1/(n+1) from the start, or prove the conjecture, now proved for n ≤ 14.
The Littlewood conjecture
Prove that n‖nα‖‖nβ‖ gets arbitrarily close to 0 for every pair of real numbers α and β.
Lehmer's totient problem
Decide whether some composite number n has Euler's totient φ(n) dividing n − 1, as every prime does.
Lehmer's Mahler measure problem
Beat Lehmer's 1.17628 with an integer polynomial of smaller Mahler measure above 1, or prove the measures stay away from 1.
The irrationality measure of pi
Lower the proven upper bound 7.10186 on the irrationality measure of π, which is conjectured to equal 2.
Integer distances in general position
Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
Imaginary quadratic fields with a prescribed class number
Give an algorithm that, for h between 2^90 and 2^100, quickly finds an imaginary quadratic field with class number exactly h.
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.
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 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.