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Maths Extra Credit Homework Assignment - Number Theory




Question;EXTRA CREDIT HOMEWORK 1;1. In this assignment, try to solve Pell's equation forn=2:x2?2y2=1.;(1) First, nd a smallest non-trivial solution in the following way: start from;y=1, nd the smallest positive integery0such that2y2;0+1is a complete square.;Then, easy to see that;?;2y2;0+1, y0)is a solution. (You can think of(1,0),(?1,0);as trivial solutions.);(2.1) Dene an operation X:(x1, y1)X(x2, y2)=(x1x2+2y1y2, x1y2+x2y1). Ex-;plain why this is a natural denition. (Hint: what is the product ofx1+;?;2y1and;x2+;?;2y2, under usual multiplication of real numbers?);Hereafter, we denote(x, y)m=(x, y)X:::X(x, y);???????????????????????????????????????????????????????????????????????????????????????????????????;mcopies;form>0. Whenm<0, and;(x, y)is an integer solution to the equation, we denote(x, y)m=(x, y)?1X:::X(x, y)?1;???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????;mcopies;where(x, y)?1=(x,?y). Dene(x, y)0=(1,0).;(2.2) Explain why(x, y)?1=(x,?y)is a natural denition. (Hint: If(x, y)is a;solution, what is1;x+;?;2y;?);(3)Show that if(x, y)is an integer solution to the equation, so are?(x, y)m, for;anym, where?(a, b)is just the pair(?a,?b). (Hint: if(x+;?;2y)(x?;?;2y)=1;then(x+;?;2y)m(x?;?;2y)m=1.) Similarly, if(x, y)and(x., y.)are solutions, so;is(x, y)X(x., y.).;Now, let(x0, y0)be the solution you found in part (1), show that all solutions are;of the form?(x0, y0)min the following way;(4.1) Show that if there is a solution not of the given form, then there is a pair of;positive integers(x., y.)which is a solution, such that(x0+;?;2y0)M y., orx


Paper#60853 | Written in 18-Jul-2015

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