Hatcher-Topology-of-Numbers · Problem 8

The Farey Diagram

Mathematics — Algebraic Topology · rational-points-on-circle

Problem

(a) Derive formulas that give all the rational points on the circle x2+y2=2x^2 + y^2 = 2 in terms of a rational parameter mm, the slope of the line through the point (1,1)(1, 1) on the circle. (The value m=∞m = \infty should be allowed as well, yielding the point (1,−1)(1, -1).) The calculations may be a little messy, but they eventually simplify to give formulas that are not too complicated: x=m2−2m−1m2+1y=−m2−2m+1m2+1x = \frac{m^2 - 2m - 1}{m^2 + 1} \qquad y = \frac{-m^2 - 2m + 1}{m^2 + 1} (b) Using these formulas, find five different rational points on the circle in the first quadrant, and hence five solutions of a2+b2=2c2a^2 + b^2 = 2c^2 with positive integers a,b,ca, b, c. (c) The equation a2+b2=2c2a^2 + b^2 = 2c^2 can be rewritten as c2=12(a2+b2)c^2 = \frac{1}{2}(a^2 + b^2), which says that c2c^2 is the average of a2a^2 and b2b^2, or in other words, the squares a2,c2,b2a^2, c^2, b^2 form an arithmetic progression. One can assume a<ba < b by switching aa and bb if necessary. Find four such arithmetic progressions of three increasing squares where in each case the three numbers have no common divisors.

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