Comparison of Known Method with the New Diophantine Method (Part IC)

A known method for obtaining Pythagorean triples is described in this section. The method is applicable for Pythagorean triples but not for the general Diophantine equations described in Part I through Part VIII. This section can be used for comparison purposes with the new Diophantine method of Parts I-VIII where the same n values are used.

The values for b and c were computed as follows and are used in the headers of columns 3 and 4 of tables A thru D:

Value for b
a² + b² = c²
c = b + n
a² + b² = (b + n)²
= b² + 2bn + n²
b = (a² − n²) ∕ 2n

Value for c
a² + b² = c²
b = c − n
a² + (c − n)² = c²
c² = a² + c² − 2cn + n²
c = (a² + n²) ∕ 2n

Tables of Pythagorean Triples Using the Natural Numbers as Values for a

This known method using just the natural numbers for the value of a produces both primitive and non primitive right angled triangles. By primitive is meant that there are no common factors among the three numbers, unlike the non primitive which do have common factors. The method also produces non integral values (fractions or integer) which can be converted to primitive form by multiplication by the n value for that particular table but if not possible by a number that can convert the triples into primitive right angled triangles either by division or multiplication. Although the new Diophantine method produces primitive as well as non primitive forms, all the values are integers unlike the regular method described herein. While the values for the headers in Part I are just the regular values of tables A thru D in this section multiplied by n with columns 1 and 2 swapping, the values of Parts II thru VIII are newly derived.

Employing the natural numbers for the values of a, Tables A thru D were produced. The tables follow the format, a row of non primitive value followed by reduction to their primitive forms if not already in their primitive forms. The first value of the table corresponds to Sample 1 where the value of a is equal to the value of n where n is a natural number, resulting in a value of zero for b in column three (the value of b = (a² − n²) ∕ 2n as calculated above). Using this particular table format ensures that no negative values for a or b are generated.

Table A
Sample	a	(a² − 1) ∕ 2	(a² + 1) ∕ 2
1	1	0	1
2	2	1.5	2.5
3	4	3	5
	3	4	5
4	4	7.5	8.5
	8	15	17
5	5	12	13
	6	17.5	18.5
6	12	35	37

Table B
Sample	a	(a² − 4) ∕ 4	(a² + 4) ∕ 4
1	2	0	2
2	3	1.25	3.25
	12	5	13
3	4	3	5
4	5	5.25	7.25
	20	21	29
5	6	8	10
	3	4	5
6	7	11.25	13.25
	28	45	53

Table C
Sample	a	(a² −9) ∕ 6	(a² + 9) ∕ 6
1	3	0	3
2	4	1¹⁄₆	4¹⁄₆
	24	7	25
3	5	2²⁄₃	5²⁄₃
	15	8	17
4	6	4.5	7.5
	4	3	5
5	7	6²⁄₃	9²⁄₃
	21	20	29
6	8	9¹⁄₆	12¹⁄₆
	48	55	73

Table D
Samplen	a	(a² − 16) ∕ 8	(a² + 16) ∕ 8
1	4	0	4
2	5	1¹⁄₈	5¹⁄₈
	40	9	41
3	6	2.5	6.5
	12	5	13
4	7	4¹⁄₈	8¹⁄₈
	56	33	65
5	8	6	10
	4	3	5
6	9	8¹⁄₈	12¹⁄₈
	72	65	97

This concludes Part IC.
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