Pythagorean n-ples and Their Equivalence for Parameterizations
Explore the concepts of Pythagorean triples, 4-tuples, and their parameterizations in mathematics. Learn about the rational points on the unit circle and sphere, derived intersections, and the three-dimensional case of Pythagorean 4-tuples. Discover key equivalences through rational points and parallel vectors in this insightful study inspired by Manjul Bhargava’s research.
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Pythagorean n-ples Dan Kalman American University Slides At: www.dankalman.net Spring SE Section Meeting 2/28/2025
Pythagorean Triples Equivalent Defs: Integer triple (?,?,?) for which ?2+ ?2= ?2 ? ?,? ?,? is integer vector with integer length ?. is rational pt on unit circle ? Parameterization Take ? > ? + (?2 ?2,2?? ,?2+ ?2) is always a PT Example: ?,? = 3,2 ?,?,? = (5,12,13)
Pythagorean 4tuples (P4Ts) Equivalent Defs: Integer 4tuple (?,?,?,?) for which ?2+ ?2+ ?2= ?2 ? ?,? ?,?,? is integer vector with integer length ?. ?,? ?is rational pt on unit sphere Parameterization Params ?,?,? give rise to P4T ?2 ?2 ?2,2??,2??,?2+ ?2+ ?2 Example: ?,?,? = 3,1,2 ?,?,?,? = (4,6,12,14) 16 + 36 + 144 = 196 = 142
Compare Parameterizations PT: ?2 ?2,2?? ,?2+ ?2 P4T:( ?2 ?2 ?2,2?? ,2?? ,?2+ ?2+ ?2) P5T: ( ?2 ?2 ?2 ?2,2??,2??,2??,?2+ ?2+ ?2+ ?2) P?T: ?1,?2, ,??where ?1,?2, ,?? 1 , and ?1= ?1 ??= 2?1??for ? = 2 ? 1 ??= ?1 ? 1?? 2 2 ?=2 ? 1?? 2 2+ ?=2
Derivation Inspired by Manjul Bhargava s 2011 Hedricks Lecture derivation for PTs Characterizes rational points on unit circle as intersections of the circle with lines If a line has rational slope, and passes through one rational point on the circle, the other intersection will also be a rational point This argument extends perfectly to intersections of lines with the unit sphere in n dimensions
Three Dimensional Case ? ?,? ?,? ?,?,?,? is a P4T iff on unit sphere in 3. Consider a line through ( 1,0,0) parallel to a rational vector, and hence to an integer vector ?,?,? Assume ? 0 so the line is not tangent to the sphere We find the other point of intersection, show it is a rational point ? is a rational point
Key Equivalence Let ? be a line through ? = ( 1,0,0) not tangent to the sphere. There is a point ? ? where ? meets the sphere Claim: ? is a rational point iff ? is parallel to a rational vector (iff parallel to an integer vector) ( ) Assume ? is a rational point. Know ? is a rational point. ? ? is rational and ? ? ? ( ) Assume ? ? for some integer vector ? . Then ? = ? + ?? ? . Next slide: Intersect with sphere to find ?, show it is a rational point.
Solving for ? Line ?: all points ? ? = ? + ?? ? not tangent to sphere so ? ? 0 ? on unit sphere iff ? ? =1 ? + ?? ? + ?? = 1 ? ? + 2?? ? + ?2? ? = 1 But ? = 1,0,0 ? ? = 1 2?? ? + ?2? ? = 0 ?(2? ?+?? ?) = 0 ? = 0 or ? = 2? ? ? ? ? = ? 0 ;Q = ? ?1 ?1
Finding Rational Points Line ? through ? parallel to the integer vector ? = (?,?,?) ? not tangent to the sphere at ? so ? 0. Q = ? ?1 = ? + ?1? = 1,0,0 + ?1(?,?,?) Q = 1,0,0 + 2? ? ? ? ?,?,? = 1,0,0 + ? ??,?,? Q = ? ? Q = 2? 1 ? ?,0,0 + 2?2,2??,2?? 1 ?2+?2+?2?2 ?2 ?2,2??,2?? ?2 ?2 ?2,2??,2??,?2+ ?2+ ?2 is P4T
Example ? = (?,?,?)= (1,2,3) P4T is (12,4,6,14) Divide out 2 to find a primitive P4T (6,2,3,7) We also obtain all the integer multiples Also can permute first three entries (6,3,2,7), 2,6,3,7 , (6,2,3,7), etc Is there an efficient way to choose (?,?,?) s so that we obtain every possible P4T without any duplication?
As some of you may know Bhargava s derivation of (?2 ?2,2?? ,?2+ ?2) was presented in a 2011 Math Horizons article coauthored with Nathan Carter. It parodied the then popular Harry Potter books/movies. We called it Harvey Plotter and the Circle of Irrationality. Today s presentation cries out for a sequel ..
