The Structure of Primitive Pythagorean Progressions

The Primitive Pythagorean triples (a^2+b^2=c^2) have a structure such that c is always odd and, if a is odd, then b is even, or vice versa (see http://mathworld.wolfram.com/PythagoreanTriple.html ). It is suggested that this structure is related to the fact that odd squares are of the form 4n+1 and even squares are of the form 4n. In addition, it is also suggested that the structure for other primitive Pythagorean progressions (e.g., a^2+b^2+c^2=d^2, a^2+b^2+c^2+d^2=e^2, a^2+b^2+c^2+d^2+e^2=f^2, etc.) must be in accordance with these two forms.

The implication of this is that, in primitive Pythagorean progressions, odd squares on the left hand side of the equation must occur in the quantities 1, 4, 5, 8, 9, etc. (4n+1 and 4n) if there are to be integer solutions. The right hand square will be odd if the number of odd squares is odd (1,5,9, etc.), or even if the number of odd squares are even (4,8, etc.).

For example, consider the primitive Pythagorean progression, a^2+b^2+c^2=d^2. If the above is correct, only one of the three left hand terms (a,b,c) can be odd, while the other two must be even and d will be odd, if there are to be integer solutions.

In the primitive Pythagorean progression, a^2+b^2+c^2+d^2=e^2, there are two possible scenarios:

1) one of the four left hand terms (a,b,c,d) can be odd, while the others three must be even and e will be odd, if there are to be integer solutions
or
2) all four of the left hand terms are odd and e is even, if there are to be integer solutions

In the primitive Pythagorean progression, a^2+b^2+c^2+d^2+e^2=f^2,
there are two possible scenarios:

1) one of the five left hand terms (a,b,c,d,e) can be odd, while the others four must be even and f will be odd, if there are to be integer solutions
or
2) four of the five left hand terms (a,b,c,d,e) must be odd, while the other one must be even and f will be even, if there are to be integer solutions

At first glance it would also appear that all five left hand terms could be odd, with f being odd. However, it is thought that the second scenario precludes this (and, it is suggested that this would be the case at 9, 13, 17, etc., left hand terms). However, five odd left hand terms with a sixth (or more) even term can produce an odd seventh term (e.g., 1^2+3^2+5^2+7^2+9^2+82^2=83^2).

etc.