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Brocard's conjecture states that pi(p_(n+1)^2)-pi(p_n^2)>=4 for n>=2, where pi(n) is the prime counting function and p_n is the nth prime. For n=1, 2, ..., the first few ...

Define g(k) as the quantity appearing in Waring's problem, then Euler conjectured that g(k)=2^k+|_(3/2)^k_|-2, where |_x_| is the floor function.

Grimm conjectured that if n+1, n+2, ..., n+k are all composite numbers, then there are distinct primes p_(i_j) such that p_(i_j)|(n+j) for 1<=j<=k.

The Hodge conjecture asserts that, for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually rational linear ...

Andrica's conjecture states that, for p_n the nth prime number, the inequality A_n=sqrt(p_(n+1))-sqrt(p_n)<1 holds, where the discrete function A_n is plotted above. The ...

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was ...

The first of the Hardy-Littlewood conjectures. The k-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, ...

The nth coefficient in the power series of a univalent function should be no greater than n. In other words, if f(z)=a_0+a_1z+a_2z^2+...+a_nz^n+... is a conformal mapping of ...

A conjecture which relates the minimal elliptic discriminant of an elliptic curve to the j-conductor. If true, it would imply Fermat's last theorem for sufficiently large ...

A conjecture which treats the heights of points relative to a canonical class of a curve defined over the integers.