whose short legs differ by one.

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: **x ^{2} + y^{2} =
z^{2}** 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.

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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