Download Advances in number theory: Proc. 3rd conf. of Canadian by Fernando Q. Gouvea, Noriko Yui PDF

By Fernando Q. Gouvea, Noriko Yui

ISBN-10: 0198536682

ISBN-13: 9780198536680

This e-book comprises the complaints of the 3rd convention of the Canadian quantity conception organization. The 38 technical papers offered during this quantity speak about proper and well timed concerns within the fields of analytic quantity thought, arithmetical algebraic geometry, and diophantine approximation. The e-book contains a number of papers honoring Paulo Ribenboim, to whom this convention was once devoted.

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Extra resources for Advances in number theory: Proc. 3rd conf. of Canadian Number Theory Association, 1991

Sample text

For this, we set x = y + c, where y is a new unknown and c is some number. Substituting this value for x in our equation, we obtain a term of the form am(y + c)m from each term amxm . We represent each such new term as a polynomial in y according to the binomial formula and then combine like terms. As a result, we obtain a new polynomial in y, which is denoted by g(y) = f(y + c). Because y is expressed in terms of x (y = x - c), the equations f (x) = 0 and g(y) = 0 are equivalent: a root y = a - c of the equation g(y) = 0 corresponds to the root x = a of the equation f(x) = 0, and a root x = {3 + c of the equation f (x) = 0 corresponds to the root y = {3 of the equation g(y) = O.

Conversely, if k > (n + 1)/2, then we similarly obtain C~ < C~-l. Therefore, the numbers in one row of the Pascal triangle increase to the middle of the row and then decrease. If n is even, then we have one largest number C;:/2, and if n is odd, then we have two neighboring numbers both equal to the largest value, CAn - 1)/2 and n +l)/2. In this case where k = (n + 1)/2, we have CA k- 1 Cnk = Cn ' Formula (21) (with the values of the binomial coefficients C~ determined by formula (24)) can be given in a somewhat more general form.

Dividing x 2 + px + q by x - a, we obtain x-a x+p+a +px +q -ax (p+ a)x +q (p + a)x - a(p + a) q+pa+ a 2, that is, x 2 + px + q = (x - a)( x + p + a) + (a 2 + pa + q). Because a is a root of the equation x 2 + px + q = 0, we have a 2 +pa+q = 0 and therefore x 2 + px + q = (x - a)(x + p + a). By our definition, the equation has two roots equal to a if x+p+a has the root a, that is, 2a+p = O. Hence, a = -p/2. Because a 2 +pa+q = 0, we can substitute a = -p/2 and obtain -p2/4 + q = O. This is the well-known condition for the equation x 2 + px + q = 0 to have equal roots.

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