Question

3: The alpha curve in R is the set A = {(x,y) € Ry2 = 2* + ?}. Define f : A\{(0,0)} R {1,-1} as follows: given (0,0) + (a,b) € A let f(a,b) = u where u is the unique) real number such that the point (1, u) lies on the line through (0,0) and (a,b). Define g: R {1, -1} → A\ {(0,0)} as follows: given u ER with u +1, let g(u) = (a,b) where (a,b) (0,0) be the unique) non-zero point on A which lies on the line through (0,0) and (1, u). (a) Find a formula for f(a,b) and a formula for g(u) and show that and g are inverses. (b) Show that the points (1,y) € Q2 with y2 = x + 22 are given by (x, y) = (v2 - 1,23 – u) with u EQ. (c) Show that the points (x,y) € Z with y2 = x3 + x2 are given by (x,y) = (u2 - 1,43 - u) with u EZ. (d) Let p be a prime number. Determine the number of points (x,y) € Z,? with y= r* +2?.

just c & d

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Answer 1

Olet (ny) EZ with y ² x (2+1). Since xE2 XHEZ xv(x+1) € 2 Y Ez at Jati EZ = √x+T EZ het n+1= +1 = u?, uez .. (ay) (d) - y =u²1) u (U²1, u²u). Le to be a prime number Zp = {0, 1, ... , Ti} squaring each elenutof of Rp the set 2p weget a subset with curdinality + can findi. Pt elennt of form u²1, Bergt we hout liply to byl Hence we can & elements are 40, 1, 2, ..., for the form 43 u we have élents 20, 1, 2, ..., 1} Hanee number of points PC)