Tags » Arithmetic Progression

Vandermonde Submatrices and Arithmetic Progressions

Currently, we are asking whether all submatrices of the order- Vandermonde matrix over a finite extension of are invertible where is prime. The answer is “no” in general: there are examples of fields where the Vandermonde matrix has a singular submatrix. 863 more words

Mathematics

Prime-Generating Polynomials

If you haven’t seen it before, the polynomial seems to look like any other. And yet, as Euler noted, this polynomial has a curious property — evaluating at the integers gives a new prime each time: 862 more words

Number Theory

Sums of Squares and Density

Lagrange’s Four Square Theorem (Lagrange, 1770) is the well-known result that every positive integer can be written as the sum of four integer squares. This was strengthened by Legendre’s 1797-1798 proof of the similar-sounding… 1,275 more words

Analytic Number Theory

Arithmetic Progression OCR AS Level May 2009

The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44.

(i) Find the first term and the common difference. 86 more words

AS Level

Units Groups and the Infinitude of Primes

Throughout, we take as a complex subring (with unity). ┬áIn this article, we’ll be interested in natural analogues of Euclid’s proof of the infinitude of the primes (i.e. 1,157 more words

Number Theory