The Diophantine Equation x² + 2y² = z² (Part III)

A Method for Generating the Variables

Squares of Magic Squares Equation

Andrew Bremner's article on squares of squares included the 3x3 square:

The numbers in the right diagonal as the triple (205²,425²,565²) seem to have been derived elsewhere. This sequence as well as a host of others has been obtained from a number of web pages starting with PartIA-1 where the first number in the triple when added to a difference (Δ) gives the second square in the triple and when this same (Δ) is added to the second square produces a third square. In other words adding the two sums produces the Diophantine equation 2y² − x² = z². The method discovered therefore was useful in generating integers which could be used as the right diagonal entries of a magic square consisting primarily of squares. These integers, as well as a host of others, became entries of this particular Diophantine equation as was shown in Part I and Part II.

The Diophantine equation x² + 2y² = z² is well known and only a small number of values are available for this equation. This web page will show how to calculate many other triples which satisfy this equation using a simple novel method, completely different from the method used in Part I and Part II for the Diophantine equation 2y² − x² = z². The method has already been covered in Part IA, Part IB, Part IC, Part IIC, Part IIIC, Part ID and Part IE, but its relation to x² + 2y² = z² was not shown until now even though several of the triples found via this method were used to construct eight magic squares containing seven squares each.

Generation of Tables for odd and even numbers

There are two slightly different methods for construction of the triples. I will cover only the preferred method shown below since it is simple, consistent and captures all the triples for that particular series. The other method is covered in Tables VII and VIII of Part IB. The preferred method uses tables of odd and even initial numbers starting first with Table I and Table II, where we use 1 for the first odd number and 2 for the first even number, then with Tables III to IX which covers other odd and even numbers. The triples are tabulated using the following titles for odd and even numbers:

where k is the increment starting at 0 and j equals the odd or even number chosen from the two sequences listed just below, ai and c equals the equations just below them in the table above; and where b equals (c² − a²)/2 in terms of k. In addition, the δ₁i and δ₂ are the differences between the ai and c columns, respectively.

What are the initial numbers that may be used in the tables? By initial I mean the ns in the triple (-ni, 0, n). The tables of initial even numbers are generated only from those numbers in the sequences generated from 2n², viz., 2,8,18,32,50,72,98,128,162,200... while the table of initial odd numbers are generated from (2n + 1)², viz., 1², 3², 5² 7², 9²... Only those numbers in these two sequences serve as the initial numbers as well as satisfy the Diophantine equation −xi² + 2y² = z² where −xi² = x².

The next two tables III and IV, are shown below and use even number 8 and odd number 9 from the two sequences above where the light green triples of Table III are divisible by 4 and the light blue of Table IV by 9. In addition, the factored triples of III and IV are contained in previous tables.

Tables V, VI and VII tabulate the results of integers 18, 25 and 32, respectively.

Tables VIII and IX tabulate the results of integers 49 and 50, respectively.

Only nine integers were used in the calculation for the Diophantine equation x² + 2y² = z² to get an idea of its versatility. Note that the tables produced can be expanded downward indefinitely and that the entries in the two sequences, used as the initial numbers, are infinite in number, thereby, giving rise to an infinite number of triples for this Diophantine equation.

This concludes Part III. To go to Part IV which covers the Diophantine equation x² − 2y² = z². To go back to Part II.

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