13. How many different Pythagorean triples do you think there are? Explain. 14. The Pythagorean triple (3, 4, 5) is the most basic triple there is. It generates many other related triples. Find several of these triples, and explain how they are generated by the triple (3, 4, 5). 15 Prove that there are in fact infinitely many Pythagorean triples in the family generated by (3, 4, 5).3.3 Pythagorean Triples Although we generally recite the Pythagorean theorem in its algebraic garb, we are no doubt aware of its clear links to geometry. The Greeks did not have algebra as we would think of it new; they thought of results like the Pythagorean theorem in purely geometric terms. However, it is also the case that the Py'tbagoreans believed that "all is number." The connection between the Pythagorean theorem and special relationships between certain numbers was not lost on the Greeks. The numbers 3, 4, 5 are said to form a Pythagorean triple because they are positive integers that satisfy the Pythagorean theorem: 32+42 =52 since 32+42 =a+15=25=52. It is likely that you remember such triples from learning about the Pythagorean theorem. 8.3.1 Pythagorean Triples as Partitions There is a close connection of Pythagorean triples to earlier results we have been considering in this book. Namely, 32 +42 is a square partition of the square 52. Prior to this, we have been trying to nd patterns among partitions of a given type. For example, in Section '47.? of Chapter 7 we showed that it is possible for every positive integer to be square partitioned by 4 of fewer squares. That's ne, but it is also interesting to consider whether there are numbers that can be partitioned in particularly nice ways. Partitioning 52 as 12 + 12 + . .. + 12 is quite boring any number can be square partitioned in such a way. In discovering Waring's problem previously, we looked at the minimum partitions. The number 52 has a much more interesting minimal partition =f+ This partition is the simplest possible square partition after the trivial 52 = 52. Thus, 3,4, and