Diophantine equations as a context for technology-enhanced training in conjecturing and proving

Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.

Exercise sheet 1

Exercises and solutions in LaTex

Exam 1999 pdf

Exam and solutions in PDF

Exam 2000

Exam and solutions in LaTex

Exam 1999

Exam and solutions in LaTex

Exercise sheet 1 pdf

Exercises and solutions in PDF

Exam 2000 pdf

Exam and solutions in PDF

The padding scheme for RSA signatures

The RSA scheme is used to sign messages; however, in order to avoid forgeries, a message can be padded with a fixed string of data P. De Jonge and Chaum showed in 1985 that forgeries can be constructed if the size of P (measured in bytes) is less than the size of N/3, where N is the RSA modulus. Girault and Misarsky then showed in 1997 that forgeries can be constructed if the size of P is less than the size of N/2. In 2001, Brier, Clavier, Coron and Naccache showed that forgeries can still be constructed when the size of P is less than two thirds the size of N. In this paper, we demonstrate that this padding scheme is always insecure; however, the complexity of actually finding a forgery is O(N). We then focus specifically on the next unsettled case, where P is less than 3/4 the size of N and show that finding a forgery is equivalent to solving a set of diophantine equations. While we are not able to solve these equations, this work may lead to a break-through by means of algebraic number theory techniques.

De solutione problematum diophanteorum per números integros : o primeiro trabalho de Euler sobre equações diofantinas

The present dissertation analyses Leonhard Euler´s early mathematical work as Diophantine Equations, De solutione problematum diophanteorum per números íntegros (On the solution of Diophantine problems in integers). It was published in 1738, although it had been presented to the St Petersburg Academy of Science five years earlier. Euler solves the problem of making the general second degree expression a perfect square, i.e., he seeks the whole number solutions to the equation ax2+bx+c = y2. For this purpose, he shows how to generate new solutions from those already obtained. Accordingly, he makes a succession of substitutions equating terms and eliminating variables until the problem reduces to finding the solution of the Pell Equation. Euler erroneously assigns this type of equation to Pell. He also makes a number of restrictions to the equation ax2+bx+c = y and works on several subthemes, from incomplete equations to polygonal numbers

Tópicos de teoria dos números algébricos e aplicações em reticulados e equações diofantinas

Pós-graduação em Matemática Universitária - IGCE

On the Generation of Heronian Triangles

We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+ε)). We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems.

Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0

This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].

Automata–based Method for Solving Systems of Linear Constraints in {0,1}

We consider a finite state automata based method of solving a system of linear Diophantine equations with coefficients from the set {-1,0,1} and solutions in {0,1}.