Birch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Diophantine equations
Equations over integers and rational numbers, from elliptic curves to the limits of decision algorithms.
Subjects: Number theory · Arithmetic geometry · Logic and computation
Problems in this collection
7 problemsBombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Hilbert’s tenth problem over the rationals
Can an algorithm always decide whether an equation with integer coefficients has a solution made of rational numbers?
Erdős–Straus conjecture
Can every fraction 4/n, with n at least 2, be written as the sum of three reciprocals of positive integers?
Erdős–Moser equation
Can a power of an integer equal the sum of the same powers of every smaller positive integer, apart from 1 + 2 = 3?
Brocard–Ramanujan problem
Can a factorial plus one be a perfect square beyond the three known examples?
Pillai conjecture
For each fixed positive gap, are there only finitely many distinct pairs of integer perfect powers separated by that gap?