Question

8.20 Question. Which natural mumbers can be written as the sum of two squares of natural raumbers? State and prove the mast general theorem possible about which natural numbers can be written as the sum of two suares of nutural numbers, and prove it. We give the most gencral result next. 8.21 Theorem. A natural number n can be written as a sum of two squares of natural mumbers if and only if every prime congruent to 3 modulo 4 in the unique prime factorization of n occurs to an even power Pythagorean triples revisited We are now in a position to describe the possible values for the hypotenuse in a primitive Pythagorean triple. 8.22 Theorem. f (a. b, c) is a primitive Pythagorean iriple, then c is a product of primes each of which is congruent to 1 modulo 4. 8.23 Theorem. If the natural number c is a product of primes each of which is congruent to 1 modulo 4, then there exist integers a and b such that (a.h.c) is a primitive Pythagorean triple.

Answer 1

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