The document describes the quadratic sieve algorithm for integer factorization. It provides examples of applying the algorithm to factor the integer 112373. Specifically, it finds values of x such that x^2 - 112373 is factorable into prime factors. It then uses the factorizations to deduce that 112373 is equal to 45 * 2472 + 1133, and continues further factorizing the terms on the right hand side to fully factor 112373.
This document discusses the design of a new spacecraft called TSUJIMOTTER that will launch in 2029. It will orbit the moon and use a special camera to take high resolution photos of the lunar surface. The camera can rotate 120 degrees to capture wide angle shots and will help scientists learn more about the geological history of the moon.
This document provides a 3 sentence summary of a longer text:
The document discusses speed limits and traffic stops. It notes that the speed limit was 60 mph and the car was traveling at 71 mph when stopped. A formula is given for calculating the probability of an event given the overall and conditional probabilities.
This document discusses solving quintic (degree 5) polynomial equations. It presents the general form of a quintic polynomial and methods for finding its roots, including using the factorization of polynomials and representations of the roots in terms of radicals. It also discusses representing functions as tensors and decomposing them into products of simpler tensors.
1) The document discusses representations of prime numbers as sums of squares and relates them to modular forms.
2) Several formulas are presented for expressing prime numbers as sums of squares of integers, such as p = X^2 + 7Y^2.
3) Properties of modular forms are summarized, including representations as Fourier series and transformations under substitutions.