Pythagorean Parabolas

I recently came across a note on an Annual Meeting of the Rocky Mountain Section of the MAA in 1923. Among the list of presentations was one by W. J. Hazzard, Professor at the Colorado School of Mines on the topic of "Parabolic Grouping of pythagorean triangles."
I was a little familiar with Prof. Hazard as I had leapfrogged off one of his old posts in the Mathematics Teacher on methods of solving a quadratic equation to write a little about the history of solving quadratics in twenty or so different ways, which I hope someday to reduce to blog posts, but not today. 
I even had a copy of one of the good Professor's books in my collection of old math books, but I had not read, nor was I aware of the idea he spoke of.  With a few words of guidance from a "very" brief coverage in the article, I was able to extract at least a little that may be of interest to anyone who enjoys Pythagorean relations, and especially if you teach high school math.

If you put one acute vertex of a right triangle at the origin and lay it out so that the shorter leg lies along the positive x-axis, the other vertex will be at the point (a,b) as determined by the two legs of the triangle.  In the graph I have shown the  position of a 3-4-5 triangle and a 5-12-13 triangle to make my meaning clear.

A natural question is, "So What?"  But if we look at several of the points determined by the upper vertex, and select out only some "related" Pythagorean triples, we notice a pattern.  In the image at right the points represent the set of triples 3-4-5; 5-12-13; 7-24-25; and 9-40-41.  (Any teacher or student who is not aware, there is a simple trick to find an infinite number of these triangles with a longer side one less than the hypotenuse.  Just take any odd number to be the short leg, square it, and then divide by two and round up to the next whole for the hypotenuse. ) 
All the points lie on a parabola y= 1/2 x² - 1/2 .  It is easy to see that the focus is at the origin, and if we think about the definition of a parabola as the set of points equally distant from a focus and directrix, we realize the line of the directrix must be the line y = -1 so that, for instance the point (3,4) which is 5 units from the origin will also be 5 units away from the directrix.

Admittedly that is a pretty small (although infinity large) sub-set of the Pythagorean triples.  What would happen if we plotted other triangles like 8-15-17?  It turns out they are not on the parabola drawn... they are on another one.  In fact, all the triangles which have a longer leg two less than the hypotenuse will also have a focus at the origin, and the directrix will be ... yeah you knew it would be, y = - 2.  You can write the equation with ease for the parabola passing through any of these Pythagorean vertices, and all the ones with a common difference between the longer leg and hypotenuse share a parabola.

All the triples I've picked so far have been primitive triples.  A good question to ask is what would happen if we picked, say, a 6-8-10 triangle. Will it fall on the same parabola as the 3-4-5, or on the ones with a difference of two?

The image below shows parabolas for differences of 1, 2, and 8 between the longer side and hypotenuse, and point D is the 6-8-10 triangle, right there with 8-15-17 and others like it. 



I'm not sure you can swap this information for bread or ale at the local inn, but it's pretty interesting stuff.

Pandigital Primes

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. In base ten such a number might be 123456789098765444321.  If the number is prime, which is really cool, it is called a pandigital prime.  And if it uses the digits exactly once each, which is even cooler, .... Unfortunately, in base ten,  which is where a lot of us hang out the most, you can't have such a number.  Any ordering of 1,2,3,4,5,6,7,8,9,and 0 will be divisible by three, and hence - NOT prime.  Even if you leave out the zero, you can't make one with the first nine digits either for the same reason.( I know..."Ahhhhh".)

So there are a couple of ways to adjust.  We can look for primes that are n digits long and use the first n numerals, for example 2143 is a four digit prime using the numerals 1,2,3, and 4.  The problem with this approach is that there are only two of them.  The four digit one is shown, and a seven digit one is 7652413 . Any number made up of the first  2, 3, 5, 6, 8, 9, or ten digits will be divisible by three.

That leaves a couple of options.  I got started thinking about these when I was wrote a blog awhile back called "The Game of Primes ."  The object was to create a string of primes by starting with one prime number and then adding a digit each time to make the string a longer prime, but using any of the ten decimal numerals as a digit.  So you could start with 2, then add 3 to get 23, etc.  I only got to seven, you may be able to do better.  There is a nine digit prime (several of them) that has no repeated numerals.  I found 576849103 is prime  and  so is 987654103.  Having pretty much reached the ends of my manual calculating limits, I asked on twitter, "Is there a nine digit prime using distinct digits that includes a two?"
Faster than a nano-bullet I got a response from jomo ‏ @n0m0 who advised me that "First nine digit prime with distinct digits that includes number 2 is 102345689, second is 102345697 and so on again!"  

Realizing I had a computation wizard on the line (at least relative to me) I wondered aloud, (or A tweet) 

"Is it possible to form an eleven digit Pandigital Prime (ie repeating only one of the 0-9)"  Again at something akin to the speed of light,  he responded with two examples; "First pandigital prime with 11 digits is 10123457689, next one is 10123465789 and so on..." 

Then, realizing he had a rube on line whose non-programming nose could easily be pushed in the mud, he sent me a list of several... you can count them, and lock this away if you are looking for 11-digit primes... 

Here is the first few, but the whole list he has graciously placed here.  List of 11 digit pandigital primes filtered from ~10 million primes

10123457689
10123465789
10123465897
10123485679
10123485769
10123496857
10123547869
10123548679
10123568947
10123578649
10123586947
10123598467
10123654789
10123684759
10123685749
10123694857
10123746859
10123784569
10123846597
10123849657
10123854679
10123876549
10123945687
10123956487
10123965847
10123984657
10124356789
10124358697
10124365879
10124365987

The Game of Primes

I just read a post called "Math Notes" by Greg Ross at Futility Closet, and as always, it was very interesting, read it all.
But one part in particular got my mind turning. He wrote:
1
19
197
1979
19793
197933
1979339
19793393 and
197933933 are all prime.
It struck me that this could be a great student game/project.

Player one picks a number, a single prime digit.
Player two must pick a second digit so that the two form a two digit prime.
Play continues until one can not make a prime.

An interesting alternative rule cold be to allow the subsequent numbers to be added at either the front or the end of the string.

It might also be interesting to create a "map" of the strings possible..


For example the one above would be on a mapping that starts with one and then branches to 11, 13, 17, and 19 The 11 could go to 113 but then would be a dead end as 1131, 1133, 1135, and 1137 (and 1139) are all factorable. The 13 node can be extended to 131, 137, 139.
Students might amaze themselves with the long strings they could create.

For the string that started all this above, one might add 1979339333 is also prime, but if you wished, you could use 1979339339 which is prime as well.  That's a quick ten digit string of primes.  It is not easy for students to test if either of those is the end of the string, but along the way they have learned something about testing primes.

A good solitare version might be to start with the digits 0,1,2,3,4,5,6,7,8,9 and try to build a tree that leads to a ten digit prime with all the numerals by adding one digit at a time.

Three Times the Symmetry

Just saw a post at Futility Closet that makes me wonder, yet again, where does he find these things... O. K. so this one was from Scripta Mathematica (1955..what, you don't browse old math mags on a daily basis???)

If you haven't discovered his blog yet, just go there and ramble around... there is no particular theme other than, in the words of an old song, "things that make you go, Hmmm!"

The thing he posted this time was this...in layers.
First, there was this palindrome... if you don't know the word, it means it reads the same front to back and back to front, like "Madam I am Adam." only numerical. (Long after I first wrote this, I learned that the Catalan term for a numeric palindrome is "capicua", which I am told means "heads and tails.")


0264 + 4125 + 5610 = 0165 + 5214 + 4620

Ok, admit it, that's cute... not mind bending cute, but cute... but then he adds,

If you put multiplication signs in the middle of each of the terms... it is STILL true (do check please)

02 × 64 + 41 × 25 + 56 × 10 = 01 × 65 + 52 × 14 + 46 × 20

Ok, that elevates it to damn near mind bending cute... if you doubt it, go find a second example. Do it with just two four digit numbers to make it easy, or do it with five if you think that is easy...and if you manage, just to torque your brain, see if it also works if you replace the multiplication sign with a Plus sign... Yeah...

02 + 64 + 41 + 25 + 56 + 10 = 01 + 65 + 52 + 14 + 46 + 20

Nah, that's it, there couldn't be anymore... I mean what else could you do ?
No, don't even imagine that if you squared each term it would still work... don't check, that couldn't possibly be true...


No really, that would just be impossible...


Stop looking down here...you know it couldn't be true...



Stop I say

Well, I warned you... now you have only yourself to blame...