Pythagorean Triple Generators - Part III

 

 

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Two prior posts were made about Pythagorean Triples and an algorithm for producing triples (Part I, Part II).

I recently came across two additional algorithms for producing Pythagorean triples.

For reference my earlier post gave the first algorithm:

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

Below is another method.

Start with any odd number and square it.
Example: start with 3; 3x3 = 9

Next, find the two consecutive numbers that add up to the square of the starting number.

In this example, 4 + 5 = 9

The starting number, 3, and the two consecutive numbers, 4 and 5, will be a Pythagorean Triple (3,4,5).

Here’s another example:

Start with the odd number, 5. Square it: 25

Find the two consecutive numbers that add up to 25: 12 and 13

The Pythagorean Triple is (5, 12, 13)

Next is a third method which starts with any multiple of 4.

Example: 8
Take half of the starting number, 8 --> 4

Square this result, 4x4 = 16

Finally, use the two odd numbers immediately before and after this square: 15, 17

These two odd numbers and the original starting number will be a Pythagorean Triple (8, 15, 17)

One may use this method with even starting numbers that are not multiples of 4; however, the Pythagorean Triples produced can be reduced by a common denominator and will be found to be similar to a primitive triple. A primitive triple is one that cannot be reduced. For example, starting with 6; square half of 6 to get 3x3 = 9; use the numbers immediately before and after to get 8 and 10. The triple (6,8,10) is similar to the triangle (3,4,5). So, if you are trying to produce primitive triples, starting with multiples of 4, one will get triples that are irreducible triples. 

 

Here is a link to a spreadsheet I made for these different Pythagorean Triple generators: Pythagorean generator.  

Pythagorean Triples can also be found in the Online Encyclopedia of Integer Sequences: A001844 - OEIS

Reference: I read about these two additional Pythagorean Triple generators in the book Geometry for dummies by Mark Ryan. 

The site Math is Fun has good illustrations and further information on triples: Pythagorean Triples - Advanced (mathsisfun.com)

Additional formulae for producing triples are found on Wikipedia.

More interesting facts about Pythagorean Triples on Dr Ron Knott's archive.

 

Lucky Numbers

 

(Image: Icon Finder - Alpar-Etele Meder)

Lucky numbers are usually associated with people’s association of a number with something like a birthday, a sport’s jersey number, or a superstition about a particular number. There is another definition of lucky numbers. Mathematician Stanislaw Ulam defined a sequence he described as “lucky numbers.” (A prior post featured Ulam: Math Vacation: Spirals of Prime Numbers (jamesmacmath.blogspot.com))

The sequence is produced using a sieve in which one begins with the natural numbers and this list is reduced by eliminating numbers using a set of rules. The numbers remaining are termed the lucky numbers. (Another numerical sieve is the sieve of Eratosthenes, which produces prime numbers.)

The first element of the sequence is 1, the first member of the sequence of natural numbers. Next, in the first application of the sieve, every second member of the sequence is removed. This eliminates all the even numbers. The remaining sequence is 1, 3, 5, 7, 9, 11…The next surviving, or “lucky,” element is 3. Next, the sieve removes every third member of the remaining sequence.

Removing every third member leaves us with 1, 3, 5, 7, 9, 11, 13, 15, 17, 19…

The next surviving number is 7, so now every seventh member of the remaining sequence is eliminated, so we have: 1, 3, 7, 9, 13, 15, 19…

Members of the sequence under 100 are: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99

The lucky number sequence has been studied extensively and has been found to be similar to the prime number sequence.

•      Many, but not all, of the lucky number sequence’s members are prime.
•       In both sequences, the pattern is irregular.
•       As with the primes, the lucky numbers can appear as twins. Examples in the primes are: (5,7), (17,19), (41, 43). Examples in the lucky number sequence are: (7,9), (13,15), (31,33). Twin primes are discussed in the post: Math Vacation: Prime Numbers - a property rediscovered (jamesmacmath.blogspot.com)
•       The frequency of prime numbers and lucky numbers are similar.
•       The frequency of twins is similar in the prime and lucky sequences. A table comparing these frequencies is found here: MATHEWS: Lucky Numbers (archive.org)
• There exists a conjecture for the lucky numbers that is analogous to the Goldbach Conjecture for primes. That is, every even number can be expressed as the sum of two lucky numbers. 

The lucky numbers are sequence A000959 is the On-Line Encyclopedia of Integer Sequences.

A listing of the first 200,00 members of this sequence can be found here:  oeis.org/A000959/b000959.txt

 

 

Counting Sheep or Sums of Primes

 

(Image: Creative Commons — Attribution-ShareAlike 3.0 Unported — CC BY-SA 3.0 Washaweb on Iconfinder)

The other night, I had trouble getting to sleep. Instead of counting sheep, I challenged myself to adding the first n primes. The first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Therefore, for the n=1 through 10, the sums of the first n primes are: 2, 5, 10, 17, 28, 41, 58, 77, 100, 129. 

I was interested in any pattern exhibited by the sequence. After n=10, I became less confident in my ability to add late at night and then a comforting thought hit me. I realized one could easily research if others had explored this sequence by going to the website of The On-Line Encyclopedia of Integer Sequences® (OEIS®). This site was written about in a prior post: Math Vacation: What is the next number in the sequence...? (jamesmacmath.blogspot.com).

With the thought that my inquiry into the sequence could easily be answered, I fell asleep. Mission Accomplished.

The next morning, I simply entered 2, 5, 20, 17 into the site’s search bar and the following answer came up: A007504 - OEIS, The sum of the first n primes. With this sequence are links also related to sequences and references to those who have researched the sequence.

I invite readers to explore this rich resource at OEIS.

 

Book Review: A divine language : learning algebra, geometry, and calculus at the edge of old age.

Alec Wilkinson, staff writer for The New Yorker magazine, recently wrote about his quest to study math later in life. Here is a link to his September 6, 2022, article: How Mathematics Changed Me | The New Yorker. He started his quest five years ago at age 65. He also writes about this in his book, A divine language : learning algebra, geometry, and calculus at the edge of old age. I have the book on order and look forward to reviewing it.

From his article, I understand much of what Wilkinson learned during his quest as I started this math blog late in my life. This blog started as a Covid project two years ago when I was 61. We both learned strangeness of different types of infinities, mystery of design of our world, and how God in unknowable.

11/8/2022 Update - my local library got this book, and I just finished it. Wilkinson included stories about many of my favorite mathematicians. I liked that he was able to interview Chris Ferguson, who won $1,000,000 in the World Series of Poker in 2000, shortly after earning his PhD from UCLA. 

Book Review: We Have No Idea, A Guide to the Unknown Universe by Jorge Cham and Daniel Whiteson

 

(Jacket design by Jorge Cham, Penguin Random House Riverhead Books)

If you are looking for an entertaining book on cosmology, We Have No Idea fits the bill. Authors Cham and Whiteson begin the book with a few chapters on what makes up the universe where we learn that it consists mostly of dark matter and dark energy. Fitting for the title of the book, scientists still know little about 95% of the universe.

To make the material easier to read, the book is generously illustrated by Cham with cartoon-style drawings.

Footnotes include science humor topics. For example, footnote 68 refers readers to the website: Has the Large Hadron Collider destroyed the world yet?

While providing a humorous example, I found their explanation of the difference of philosophical and scientific theories to be very easy to understand. Essentially, theories need to be testable to be deemed scientific.

 Another topic explored is string theory. Those who like the post, The Very Samll and the Very Big will like the chapter "How Many Dimensions are There?" 

Book’s website: We Have No Idea by Jorge Cham, Daniel Whiteson: 9780735211520 | PenguinRandomHouse.com: Books