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§3. Pythagoras, Diophantus, and Fermat 7

There is a more elegant and general way of stating the theorem which is due to
Hasse in its final form and uses p-adic numbers.
(2.5) Hasse–Minkowski Theorem. A homogeneous quadratic equation in several
variables is solvable by rational numbers, not all zero, if and only if it is solvable
in the p-adic numbers for each prime p including the infinite prime. The p-adic
numbers at the infinite prime are the real numbers.
From this result the theorem of Legendre about the congruence follows in a very
elementary way. The p-adic theorem is the better statement, and for the interested
reader a proof can be found in Chapter 4 of J.-P. Serre, Course in Arithmetic,or
in Appendix 3 of Milnor and Husem¨ oller, Symmetric Bilinear Forms (both from
Springer-Verlag).

§3. Pythagoras, Diophantus, and Fermat

2
2
2
2
2
The simple conic with equation x + y = 1or x + y = w has a long history
stretching back to Pythagoras in the sixth century B.C. It started with the relation
between the lengths of the three sides of a right triangle

2
2
2
The relation c = a + b is attributed to Pythagoras, but it seems to have been
known in Babylon at the time of Hammurabi and to the Egyptians, besides to the
members of Pythagoras’ school in Cortona in southern Italy.
2
2
2
Triples of whole numbers (a, b, c) satisfying c = a +b are called Pythagorean
triples. Some of the first examples known from the time of Pythagoras were (3, 4, 5),
and (5, 12, 13), and (9, 40, 41). Of course, if (a, b, c) is a Pythagorean triple, then
so is (ka, kb, kc) for any whole number k. Thus it suffices to determine primitive
Pythagorean triples where the greatest common divisor of a, b, and c is 1. The above
examples are primitive. The determination of all primitive Pythagorean triples goes
back to Diophantus of Alexandria, about 250 A.D.
(3.1) Theorem. Let m and n be two relatively prime natural numbers such that n−m
2
2
2
2
is positive and odd. Then (n −m , 2mn, n +m ) is a primitive Pythagorean triple,
and each primitive Pythagorean triple arises in this way for some m, n.
This theorem follows from the considerations of the previous section where a
2
2
conic was projected onto a line in (2.3). Consider the conic x + y = 1. Project
from the point (−1, 0) the points on this circle onto the y-axis. The line L t through