Download More on Pythagorean Triples (1/29)

More on Pythagorean Triples (1/29)
• First, some True-False Questions (please click A for True
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and B for False):
(9, 14, 17) is a Pythagorean triple.
(9, 12, 15) is a Pythagorean triple.
(9, 12, 15) is a primitive Pythagorean triple.
(9, 40, 41) is a primitive Pythagorean triple.
Every pair of numbers (s, t) with s > t > 0 give rise to a
Pythagorean triple (s t, (s2 – t2) / 2, (s2 + t2) / 2).
Every pair of numbers (s, t) with s > t > 0 give rise to a
primitive Pythagorean triple (s t, (s2 – t2) / 2, (s2 + t2) / 2).
And some questions on “modulo”
• 23 and 17 are congruent modulo 11.
• 39 and 17 are congruent modulo 11.
• -5 and 17 are congruent modulo 11.
• Recall, by “a mod m” we mean the remainder upon
division of a by m. (We always assume m > 0.)
Note that this must be a number r for which 0 r < m.
• 63 mod 11 is 8
• 34 mod 7 is -1.
• If a is any integer, then there are exactly m possible
values for a mod m.
A “Non-elementary” Approach to
Pythagorean Triples
• An “elementary” solution to a number theory problem
involves staying in the realm of integers (+ and –). When
we venture outside that set, the solution becomes “nonelementary”.
• Many of the hardest problems (e.g., Fermat’s Last
Theorem) seem to require non-elementary tools.
• In Chapter 3, we look at an easy but non-elementary
approach to finding Pythagorean triples by going into the
real plane (as in pre-Calc, Calc I and Calc II).
Outline of the Method
• Starting with a Pythagorean triple a2 + b2 = c2, divide
through by c2, obtaining (a/c)2 + (b/c)2 = 1. Note that a/c
and b/c are positive proper fractions, i.e., positive proper
rational numbers, and they represent a point in the first
quadrant on the unit circle in the plane (x2 +y2 = 1).
• Note that slope m of the line connecting that point to the
point (-1, 0) is m = b2 / (a2 + c2) (work this out!).In particular,
m is itself positive, proper and rational.
• On the other hand, if we take any positive proper rational
number m, we can solve the simultaneous equations of the
circle and of the line with slope m connecting (-1, 0) to
(x, y) (i.e., y = m(x + 1)) to get a rational point on the circle.
Conclusion
• We obtain, after some algebraic effort (see text), that
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(x, y) = ((1 – m2) / (1 + m2), 2m / (1 + m2)), which must be
rational since m is rational.
Specifically, if m = v / u (where v and u are positive whole
numbers with v < u), then we get (again after algebra)
(x, y) = ((u2 – v2) / (u2 + v2), 2uv / (u2 + v2)).
Multiplying through, we get
(a, b, c) = (u2 – v2, 2uv, u2 + v2).
For example, v = 1 and u = 2, give us the triple (3, 4, 5).
Finally, connecting back to Chapter 2 (algebra again!),
s = u + v and t = u – v . Hence we get primitive PT’s
if u and v relatively prime and one is even, one odd.
Assignment for Friday
• Read Chapter 3 carefully.
• Do Exercises 3.1 a and b and 3.2.
• Had enough of Pythagoras for now? Me too. We will (after
checking this material) move on.