User:Neil Parker/Sandbox/Plimpton

Diophantus II.VIII and Plimpton 322
Diophantus II.VIII describes a technique from the Ancient World (albeit a millennia or so later than Plimpton) for generating rational triples. As is shown in the article the core algorithm - embodied in modern algebraic form -results in the generating tuple:


 * $$\frac{t^2-1}{2}; t ; \frac{t^2+1}{2}$$

This is often referred to as the ‘Platonic Sequence’ and is considered a special case (with q=1) of the Euclidian generator tuple:


 * $$p^2-q^2; 2pq; p^2+q^2\!$$

As is mentioned above the currently definitive thesis on Plimpton 322 (by Eleanor Robson) presents the entries in the ancient tablet as being derived from a set of reciprocal pairs:


 * ...the entries in the table are derived from reciprocal pairs x and 1/x, running in descending numerical order from 2;24 ∼ 0;25 to 1;48 ∼ 0;33 20 (where ∼ marks sexagesimal reciprocity). From these pairs the following “reduced triples” can be derived:


 * s = s/l = (x − 1/x)/2,
 * l = l/l = 1,
 * d = d/l = (x + 1/x)/2.

This leads to the following tuple:


 * $$\frac{t-\tfrac{1}{t}}{2}; 1 ; \frac{t+\tfrac{1}{t}}{2}$$

It should be apparent that if we scale the above tuple by a factor t, we will have the very same ‘Platonic Sequence’ arrived at by following the Diophantine algorithm.Furthermore if we substitute a rational number p/q in either the Diophantine or reciprocal pair tuples, we will – upon clearing denominators -  arrive at the Euclidian generator tuple. Euclidian co-prime numbers p,q are therefore to be found in the numerator and denominator of Robson's reciprocal pairs or for that matter in the numerator and denominator of any rational number and its reciprocal p/q and q/p respectively (p,q>0 and p>q).