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.
The document contains a series of mathematical operations combining numbers and suits from playing cards using union operations. It celebrates the discovery of the Grothendieck prime by combining the numbers 4, 6, 6, 8, 8, 8, 10, 10 with Queen and displays the result as 246810121. It also performs the calculation 10 x 13 = 1313 and combines it with King to get 1313.
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.
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.
The document contains a series of mathematical operations combining numbers and suits from playing cards using union operations. It celebrates the discovery of the Grothendieck prime by combining the numbers 4, 6, 6, 8, 8, 8, 10, 10 with Queen and displays the result as 246810121. It also performs the calculation 10 x 13 = 1313 and combines it with King to get 1313.
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.