Sum of angles of a triangle The relation between angular defect and the triangle's area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small (but positive). For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal
Fermat's theorem on sums of two squares he announced the following two results for odd primes formula_4: He also wrote: In other words, if "p, q" are of the form 20"k" + 3 or 20"k" + 7, then "pq" = "x" + 5"y". Euler later extended this to the conjecture that Both Fermat's assertion and Euler's conjecture were established by Lagrange. A generalization of Fermat's theorem, the sum of two squares theorem, characterizes the integers (not necessarily prime) that can be expressed as the sum of two squares. They are exactly the integers in which each prime that is congruent to 3 mod 4 appears with an