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Pythagorean Triples (1/24)
• Pythagorean Theorem. A triangle with sides of length
a, b and c is a right triangle if and only if a2 + b2 = c2.
• As number theorists, we are interested in this equation as
a Diophantine equation, i.e., we are interested in finding
solutions in the natural numbers. Such a (non-trivial)
solution is called a Pythagorean triple.
• In particular, if a, b and c have no common divisors, then
(a, b, c) is called a primitive Pythagorean triple (PPT).
• Questions: Are there infinitely many of these primitive
ones? Can we figure out a way to manufacture them?
Always Look at Some Data!
• Here’s are some PPT’s (in increasing order of the c’s)
• What, if anything, do they seem to have in common?
• (3, 4, 5)
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(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(20, 21, 29)
(12, 35, 37)
(9, 40, 41)
(28, 45, 53)
(11, 60, 61)
(16, 63, 65)
(33, 56, 65)
Etc.
Look, for example, at divisibility by 2 (i.e., even or odd), by
3, and by 5.
Begin Analyzing...
• So assume then that a and c are odd and b is even.
• Move b2 to the right hand side and factor that side.
Looking at examples, what seems to be true of c – b
and c + b? Can we prove that?
• Okay, so let c + b = s2 and c – b = t2 where s and t are
both odd and s > t 1. Now use algebra to show that
c = (s2 + t2) / 2 , b = (s2 – t2) / 2, and a = s t .
• So we now have a machine for manufacturing PPT’s!
Try it out!
Our First Theorem
• The Primitive Pythagorean Triple Theorem. Given any pair
of odd numbers s and t with s > t 1, we obtain a
primitive Pythagorean triple (a, b, c) with c = (s2 + t2) / 2 ,
b = (s2 – t2) / 2, and a = s t . Moreover, every PPT is
generated in this manner.
• Corollary. There are infinitely many PPT’s.
Assignment for Monday
• Read Chapter 2 very carefully and try to understand all
the math which goes on there. The arguments given are
typical of the many of the arguments in elementary
number theory. (Note: By “elementary” is not mean “easy”.
Rather, I mean doing the argument entirely within the
natural numbers, as opposed to moving out into the
rationals, the reals, or the complexes. In Chapter 3, we
shall indeed see an example of a non-elementary
approach to PPT’s.
• Do (carefully) Exercise 2.1. Hint: every number can be
written in one of the three forms 3x, 3x + 1, and 3x + 2.
• Have a good weekend, but not too good!