But this is like saying that it is interesting that two of the three number are prime because the density of prime numbers goes down as you go to larger numbers.
Or that it is interesting the 4 is a perfect square because it is so rare among integers for one to be a perfect square. After all, only 9% of the positive integers less than 100 is a perfect square while only but about 1% of the integers less than 10,000 is and only 0.1% of the integers less than one million is.
There are infinitely many Pythagorean triples (i.e., integer solutions to the Pythagorean equation). Again, plug ANY two different positive integers m and n into the formulas I gave and you have a Pythagorean triple. If m = 2 and n = 1, you have 3,4,5.
So what you are really saying is that it is somehow interesting that if you take the difference of the squares of two integers you get an integer, or that twice the product of two integers is an integer, or that the sum of the squares of two integers is an integer. But, to me, that seems pretty much like the only thing that can happen, so not very interesting.
If anything, what I would say would be somehow interesting is that there is no Pythagorean triple that consists of three prime numbers -- but it's only interesting until you look at the equations I gave you and you see that this has to be true (provided that all primitive Pythagorean triples can be generated by those equations using a suitable choice of m and n, which they can).
Or that it is interesting the 4 is a perfect square because it is so rare among integers for one to be a perfect square. After all, only 9% of the positive integers less than 100 is a perfect square while only but about 1% of the integers less than 10,000 is and only 0.1% of the integers less than one million is.
There are infinitely many Pythagorean triples (i.e., integer solutions to the Pythagorean equation). Again, plug ANY two different positive integers m and n into the formulas I gave and you have a Pythagorean triple. If m = 2 and n = 1, you have 3,4,5.
So what you are really saying is that it is somehow interesting that if you take the difference of the squares of two integers you get an integer, or that twice the product of two integers is an integer, or that the sum of the squares of two integers is an integer. But, to me, that seems pretty much like the only thing that can happen, so not very interesting.
If anything, what I would say would be somehow interesting is that there is no Pythagorean triple that consists of three prime numbers -- but it's only interesting until you look at the equations I gave you and you see that this has to be true (provided that all primitive Pythagorean triples can be generated by those equations using a suitable choice of m and n, which they can).
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