Open problems in Number Theory

Halve n if even, map n to 3n + 1 if odd, repeat. The conjecture: every positive integer eventually reaches 1.

If a complex polynomial of degree n shares a root with each of its derivatives P′, P″, …, P^(n−1), then it must be c(x − a)^n — a power of a single linear factor.

A Hadamard matrix — a ±1 matrix with pairwise orthogonal rows — exists for every order divisible by 4.

Is there a gap above 1 in Mahler measures of integer polynomials? Lehmer's degree-10 polynomial has measure ≈ 1.176280818; the question is whether any non-cyclotomic integer polynomial does better.

Does an odd number exist that equals the sum of its proper divisors? Every known perfect number is even.

Does a rectangular box exist whose three edges, three face diagonals, and space diagonal are all integers? Euler bricks (integer edges and face diagonals) exist; adding the space diagonal is the open part.