The Landau-Ramanujan constant is related to the number of integers which can be expressed as the sum of two of these numbers. A generalization of a problem involving sums of these numbers is known as Waring's problem, and the Trivial Inequality implies that all of these numbers are nonnegative. Lagrange proved that any positive integer can be written as the sum of (*) four of these numbers. These numbers are the only ones to have an odd number of factors. One of these modulo a number is known as a quadratic residue, and consecutive triangular numbers sum up to one of these numbers. The sum of the first n odd numbers is the nth one of these numbers. For ten points, name these numbers, which equal an integer times itself.
The Landau-Ramanujan constant is related to the number of integers which can be expressed as the sum of two of these numbers. A generalization of a problem involving sums of these numbers is known as Waring's problem, and the Trivial Inequality implies that all of these numbers are nonnegative. Lagrange proved that any positive integer can be written as the sum of (*) four of these numbers. These numbers are the only ones to have an odd number of factors. One of these modulo a number is known as a quadratic residue, and consecutive triangular numbers sum up to one of these numbers. The sum of the first n odd numbers is the nth one of these numbers. For ten points, name these numbers, which equal an integer times itself.
The Landau-Ramanujan constant is related to the number of integers which can be expressed as the sum of two of these numbers.