Pythagorean ratios in arithmetic progression, part II. Four Pythagorean ratios

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As in [3] let {a, b}designate the Pythagorean ratio (a² − b²)/2ab between the sides of a rational right angled triangle. The principal result of [3] is that {a, b}is the arithmetic mean of two Pythagorean ratios, and hence is the middle term of a three term arithmetic progression, if and only if a /b is the geometric mean of two Pythagorean ratios. Here in Part II we study sets of four Pythagorean ratios in arithmetic progression. We show that sets of four in consecutive places in an arithmetic progression are closely related to sets of four in the first, second, third and fifth places in a progression; any one of the former sets determines two of the latter sets, and either one of the latter sets determines the other and the former. We construct an infinite sequence of sets of four ratios in consecutive places of arithmetic progressions, the last term of each set being the first term of the next set. These sets are related to solutions of the Diophantine equations r² = 5p²q² ± 4(p⁴ − 2q⁴). Computer searches, in addition to exhibiting enough members of this sequence to enable us to identify it, also exhibited two sets which do not belong to this sequence.