Integer Pythagorean Right Triangles
whose short legs differ by one.

Last Up date: 2018 November 25 on nuMacMini /work/gd/lafarr/math
Recent changes: Put "Integer" in heading line.
This Page started on: 2007 July 04 20:52:03
Table Computed Tue 2007 July 04 05:46:02 by LaFarr Stuart

Triples below are given in x, y, z order. Hypotenuse is last.

4  3  5
20  21  29
120  119  169
696  697  985
4060  4059  5741
23660  23661  33461
137904  137903  195025
803760  803761  1136689
4684660  4684659  6625109
27304196  27304197  38613965
159140520  159140519  225058681
927538920  927538921  1311738121
5406093004  5406093003  7645370045
31509019100  31509019101  44560482149
183648021600  183648021599  259717522849
1070379110496  1070379110497  1513744654945
6238626641380  6238626641379  8822750406821
36361380737780  36361380737781  51422757785981
211929657785304  211929657785303  299713796309065
1235216565974040  1235216565974041  1746860020068409
7199369738058940  7199369738058939  10181446324101389
41961001862379596  41961001862379597  59341817924539925
244566641436218640  244566641436218639  345869461223138161
1425438846754932240  1425438846754932241  2015874949414289041
8308066439093374804  8308066439093374803  11749380235262596085
48422959787805316580  48422959787805316581  68480406462161287469
282229692287738524680  282229692287738524679  399133058537705128729
1644955193938625831496  1644955193938625831497  2326317944764069484905
9587501471344016464300  9587501471344016464299  13558774610046711780701
55880053634125472954300  55880053634125472954301  79026329715516201199301
325692820333408821261504  325692820333408821261503  460599203683050495415105

A simple formula for computing integer triples satisfying: x2 + y2 = z2 is at: pmn.htm.

At ppt.htm, I list many such triples. Looking at that page I noted there are a few "Near Isosceles" ones such as: 3,4,5 and 20,21,29 they, or mutiples of them such as 6,8,10 are useful if you are laying out a right angle. Later I was amazed when I saw 119,120,169. So. I wrote Ken Bronz asking if he thought there might be more--possibly an infinite number?

There is an infinite number of such triples! I computed those above based on the following note I got from Dr. Ken Bronz. I have no idea how Ken comes up with such magic. Working for him was a high light of the 5 years I spent working for RCA, in Cherry Hill, New Jersey.

Ken's 2007 July 03 email telling me how to generate the above table is below

Hello, again!

There ARE an infinite number of Pythagorean Right Triangles (X,Y,Z).

Given ANY positive M,N, with M > N,
  X = M*M-N*N,  Y = 2MN, Z = M*N+N*N  is a right triangle.

N = 1, 2, 5, 12, 29 ..., etc., give solutions to your problem.
    In general,  each N in the series is twice the previous plus the one
before that.
               N(J) =  2*N(J-1) + N(J-2), and M(J) = N(J+1).

Have a safe and enjoyable 4th!!!

Ken

I sent Ken an email asking if it would be OK to put his name on a web page. And got this reply. " Feel free to use my name if you wish. But YOU get the credit for finding the triangles in the first place. I enjoy working on problems. Keep them coming!!

The following was in an email from Ken's wife:
I'm sorry to tell you that, after a full day of activity last Wednesday, 2010 Feb. 10, Ken passed away in his sleep. For me a sad note. I lost my best Mathematical friend and reference.


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