From the preface to Introduction to the Construction of Class Fields ,
by Harvey Cohn (Cambridge University Press, 1985):
“It is an elementary observation that an integral right triangle
has an even area. Suppose the hypotenuse is prime.
Q. How do we determine from the prime value of the hypotenuse
when the area is divisible by 4, 8, 16, or any higher power of 2?
A. We use class fields constructed by means of transcendental
functions, of course!
The question might have been asked by Pythagoras in about
500 BC….”
The question seems to assume something apparently not known to Pythagoras:
The area is determined uniquely by the prime hypotenuse.
Nontrivial exercise: Prove or disprove this assertion.
Background to the exercise: See Fermat’s Christmas Theorem on the Web,
and a specific remark about prime hypotenuses in a letter from Fermat to
Mersenne on Christmas Day, 1640, quoted in The Mathematical Career
of Pierre de Fermat, 1601-1665 , by Michael Sean Mahoney (Princeton
University Press, 2nd ed,, 1994), pp. 316-317.
