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Pythagorean Triangles (Dover Books on Mathematics)

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A pythagorean triangle (PT) is a right triangle (i. e. a triangle which has a right angle) the lengths of the sides of which are all 'natural numbers' (i.e. positive integers) The smallest and best known PT is (4, 3, 5).

Chapter 1, section 1.2 begins: In a pythagorean triangle (as in any right triangle) the biggest side is obviously the hypoteneuse; the other two sides, called arms, contain the right angle. If these (i. e. their lengths) are x and y and the hypoteneuse is z, then by the theorem of Pythagoras,

. . x^2 + y^2 = z^2

[Sierpinski used superscripts, but Amazon's text box doesn't provide for them]

Many (I think most) Americans are used to calling the shorter sides of a right triangle legs, not arms, and we might be more comfortable with 'called' instead of 'obviously.' Also, we are used to naming the legs a and b and the hypoteneuse c, rather than x, y, and z. While differences such as these make this book require a bit more effort to read, the effort is worth it for the many interesting facts you will find here.

For example, on page 16 we learn that if the lengths of the two legs of a PT are consecutive numbers, b = a + 1, then (3a + 2c + 1, 3a + 2c + 2, 4a + 3c + 2) is another PT, On page 17 he lists the first six such triangles: (4, 3, 5), (20, 21, 29), (120, 119, 169), (696, 697, 985), (4060, 4059, 5741), and (23660, 23661, 33461). Perhaps because it is so obvious, Sierpinski doesn't mention that therefore PTs exist with acute angles arbitrarily close (but never =) to 45 degrees.

While extensive, the information about PTs in this slim volume (107 pages) is not exhaustive. In addition to the omission cited above, Sierpinski says nothing about infinite matrices of PTs, of which there are two that I know of. One is based on x=2r-1, y=2k, where r is the row number and k is the column number, and
a(r,k) = 2xy,
b(r,k) = y^2 - x^2,
c(r,k) = y^2 + x^2.
This has the advantage of formulaic simplicity, compared to:
a(r,k) = 4rk+2k(k-1),
b(r,k) = 4r(r+k-1) - 2k + 1,
c(r,k) = 4r(r+k-1) + 2k(k-1) + 1.
However, the latter gives a matrix in which every row is an infinite family of PTs in each of which c exceeds a by the square of the rth odd number (1, 9, 25, 49, . . .)
and every column is an infinite family in each of which c exceeds b by twice the square of k (2, 8, 18, 32, . . .).

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