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What can we say about x and y?
Hint: Is it possible to distinguish the "x" from the "y" in each primitive triple somehow, or are the two completely interchangeable?
Can you prove that x, y, and z have specific parities?
Hint: The methods of modular arithmetic may be useful. In particular, consider just two possible values for the variables, 0 and 1, where 1 + 1 = 0. (This is arithmetic where we only care whether a number is odd or even.)
Find generators for Pythagorean triples.
Hint: Natural choices are the simplest numbers that can represent a triple entirely. Use the regularities of z – x and z – y to define the generators r and s.
What conditions on r and s guarantee a primitive triple?
Hint: Look at pairs (r, s) that result in nonprimitive triples.
How many primitive Pythagorean triples exist?
Hint: Count the number of viable pairs of generators.
In a primitive triple (x, y, z), are any of the sides always divisible by 3, 4, or 5?
Hint: One of these numbers always divides some side; one of these numbers always divides one of the two legs; and one of these numbers always divides a specific side.
Can you prove these divisibility claims?
Hint: Use the methods of modular arithmetic.
Are there two different primitive triples that share a hypotenuse?
Hint: Find triples with hypotenuse z = 65.
Determine whether there is a number z that is the hypotenuse of exactly three/four/five triples.
Hint: A computer program (or examination of data) will tell you this. Look at the case z = 1105.
How many primitive triples have as the hypotenuse some given z?
Hint: Look at the prime factors of z.
How many (not necessarily primitive) Pythagorean triples have a given number z as a hypotenuse?
Hint: In how many ways can z be reduced (by dividing) to the hypotenuse of a primitive triple, and how many primitive triples have each reduction as a hypotenuse?
What is the nth triple satisfying z – y = 1?
Hint: Substitute y = z – 1 into
What is the nth triple that satisfies |x – y| = 1?
Hint: First find several examples. Then find some simple recurrence relations in the generators of these triples. Solve the recurrence relations explicitly to find the nth generators rn and sn, and then from these find explicit formulas for xn, yn, and zn.
What is the nth triple that satisfies |x – y| = 7?
Hint: The same procedure works as for the case |x – y| = 1, but there are two independent "threads" of the recurrence relation in the triples instead of one.
Which values are possible for |x – y|?
Hint: Look at the prime factors of the values that occur.
Find all triples with a given d = |x – y|.
Hint: Count the number of threads by factorind d over Z[√2]. Solve the recurrence relation generally.