Diophantine equations for integers.

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A Diophantine equation is any equation forwhich you are interested only in theinteger solutions to the equation.A linear Diophantine equation is a linearequation ax + by c with integercoefficients for which you are interestedonly in finding integer solutions.Math 138/Burger
Theorem 1For any nonzero integers a and b, thereexist integers x* and y* such thatgcd(a,b) ax* + by*.(Proof for Math 133!)When you have a linear Diophantine equation to solve,the first question you should ask about that Diophantineequation is whether or not the equation admits solutionsin integers.The following theorem tells you how to find the answer tothis question.
Page 1 and 2: The Euclidean Algorithm andDiophant
Page 3 and 4: Use of the gcdReducing fractionsEx.
Page 5 and 6: Euclidean Algorithm(p. 102)To find
Page 7 and 8: Theorem(3.2.2, p.105)For any nonzer
Page 9 and 10: • Solve each division problem, ex
Page 11: Ex) Now use the Euclidean Algorithm
Page 15 and 16: Theorem 4If gcd(a,b)│c, then the
Page 17 and 18: Using Division and Euclidean Algori
Page 19 and 20: • Use the Euclidean Algorithm to
Page 21 and 22: Solve 172x + 20y 10004 2 172 (1 7
Page 23 and 24: Theorem 4If the linear Diophantine
Page 25 and 26: ExampleFind all positive solutions
Page 27: Find all positive integer solutions

Diophantine equations for integers.

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