Pythagorean Triples

A Pythagorean triple is a right triangle that consists integral (whole number) sides and hypotenuse. According to Pythagoras, the sum of the squares of two perpendicular legs is equal to the square of the longest leg which is called the hypotenuse. That is,

a² + b² = c².

The well known Pythagorean triple is 3-4-5 triangle. That means the legs are 3 and 4 units and the hypotenuse is 5. The following are also known Pythagorean triples: 5-12-13, 7-24-25, 8-15-17, and many more.

As a mathematics enthusiast, I was curious how many Primitive Pythagorean triples are already discovered? Primitive Pythagorean triples are triangles with sides a, b, and c having no greatest common factor(GCF) other than 1. So, a 6-8-10 triangle is not counted because the sides are just multiples of 3-4-5, respectively.

I do not know what are the lengths of the sides of the Primitive Pythagorean triple that is already found out to be the one with the longest side. Answer.com states that it was the triangle with sides 77893200, 128189952, and 150000048 that is the largest so far.

Even though these don't make sense to other people, I had a curiosity that made me tabulate all the primitive Phytagorean triples by generating them with the use of VBA macros in excel. So far, I have generated 5003 primitive Pythagorean triples that have legs shorter than 50,000 units. I also include the VBA codes I used to generate these triangles.

These are the primitive Pythagorean triples that have hypotenuses less than 100.

No.	a-b-c	No.	a-b-c	No.	a-b-c	No.	a-b-c
1	3-4-5	5	9-40-41	9	16-63-65	13	36-77-85
2	5-12-13	6	11-60-61	10	20-21-29	14	39-80-89
3	7-24-25	7	12-35-37	11	28-45-53	15	48-55-73
4	8-15-17	8	13-84-85	12	33-56-65	16	65-72-97

Last 100 Primitive Pythagorean Triples with legs less than 50,000 units.

No.	a-b-c	No.	a-b-c
4904	43392-24095-49633	4954	45548-23325-51173
4905	43428-44525-62197	4955	45560-21462-50362
4906	43473-48464-65105	4956	45621-46060-64829
4907	43491-44620-62309	4957	45732-30115-54757
4908	43608-47705-64633	4958	45753-34504-57305
4909	43623-46864-64025	4959	45784-46287-65105
4910	43680-49319-65881	4960	45888-47905-66337
4911	43725-25172-50453	4961	45895-46728-65497
4912	43804-45453-63125	4962	45980-49539-67589
4913	43885-47532-64693	4963	46008-27206-53450
4914	43896-47047-64345	4964	46035-49028-67253
4915	43953-33896-55505	4965	46169-47400-66169
4916	43976-48657-65585	4966	46172-47235-66053
4917	43989-49820-66461	4967	46272-48865-67297
4918	44023-45936-63625	4968	46305-49712-67937
4919	44080-44799-62849	4969	46443-48076-66845
4920	44147-48204-65365	4970	46452-46565-65773
4921	44160-27387-51963	4971	46560-23236-52036
4922	44187-37516-57965	4972	46560-48191-67009
4923	44240-26718-51682	4973	46604-21597-51365
4924	44268-47995-65293	4974	46717-48756-67525
4925	44289-46600-64289	4975	46835-41652-62677
4926	44310-33496-55546	4976	46843-47124-66445
4927	44344-49617-66545	4977	46844-47517-66725
4928	44384-35550-56866	4978	46948-49155-67973
4929	44409-48880-66041	4979	46991-49440-68209
4930	44415-45008-63233	4980	47047-43896-64345
4931	44460-45731-63781	4981	47121-47800-67121
4932	44473-38136-58585	4982	47236-48477-67685
4933	44515-36612-57637	4983	47276-38493-60965
4934	44555-47268-64957	4984	47399-48480-67801
4935	44556-47333-65005	4985	47628-49445-68653
4936	44671-49560-66721	4986	47677-49164-68485
4937	44685-45668-63893	4987	47712-32825-57913
4938	44736-45073-63505	4988	47912-32175-57713
4939	44821-47940-65629	4989	47955-49852-69173
4940	44829-31220-54629	4990	48081-48200-68081
4941	44840-46671-64721	4991	48138-32600-58138
4942	44932-48285-65957	4992	48312-49735-69337
4943	45012-49915-67213	4993	48336-39010-62114
4944	45087-48616-66305	4994	48363-48884-68765
4945	45120-46009-64441	4995	48400-25359-54641
4946	45220-20928-49828	4996	48480-47399-67801
4947	45220-47619-65669	4997	48600-49049-69049
4948	45308-49245-66917	4998	48880-44409-66041
4949	45320-37791-59009	4999	49155-46948-67973
4950	45347-45396-64165	5000	49288-49335-69737
4951	45353-49296-66985	5001	49539-45980-67589
4952	45495-47672-65897	5002	49617-44344-66545
4953	45540-30749-54949	5003	49621-49980-70429

How did I generate these?

Again, I have generated the triangles with the use of VBA Macro in Microsoft Excel. The codes I used are as follows. Try to generate them even upto 1,000,000 units length. Just change the ranges of a and b in my code! :)

Are you a math enthusiast too? Comment on this so that I can follow you! :)