# Proof That There Is An Infinite Number Of Pythagorean Triples

#### Unknown Soldier

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In case you don't know, a Pythagorean triple is a set of three positive integers a, b, and c such that a^2 + b^2 = c^2. For example, if a = 3, b = 4, and c = 5, then a^2 + b^2 = c^2 = 3^2 + 4^2 = 5^2 which is true, of course, so a = 3, b = 4, and c = 5 is a Pythagorean triple.

There are many other examples of Pythagorean triples, but are there only so many of them or an infinite number of them? It seems likely that there is an infinite number of Pythagorean triples, but we mathematicians dare not rely on intuition. So here's a proof I came up with that demonstrates that Pythagorean triples are infinite in number:

Theorem: There is an infinite number of Pythagorean triples.​
Proof: Let n be any nonnegative integer. Then 2^n ∙ 3, 2^n ∙ 4, and 2^n ∙ 5 are positive integers because the set of positive integers is closed under multiplication. So using these three values we have​
(2^n ∙ 3)^2 + (2^n ∙ 4)^2 = 9(2^(2n)) + 16(2^(2n)) = 25(2^(2n)) = (2^n ∙ 5)^2.​
So by the definition of a Pythagorean triple 2^n ∙ 3, 2^n ∙ 4, and 2^n ∙ 5 constitutes a Pythagorean triple. And since the set of nonnegative integers is infinite, then there is an infinite number of Pythagorean triples of the form 2^n ∙ 3, 2^n ∙ 4, and 2^n ∙ 5 each corresponding to some nonnegative integer n. Therefore, there is an infinite number of Pythagorean triples. □​

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So according to you, one can’t prove that 2+2=4 and it’s not true anyway. OTOH, you can prove that there are an infinite number of Pythagorean triples and it’s true that there are. Interesting.

In any case, the infinitude of such triples was first proved by Euclid and the proof is trivial.

So according to you, one can’t prove that 2+2=4 and it’s not true anyway.
I never said that. What I did say is that 2 + 2 = 4 can be proved true but only by assuming arbitrary definitions and axioms.
OTOH, you can prove that there are an infinite number of Pythagorean triples and it’s true that there are. Interesting.
It's fascinating! Anyway, proving that there is an infinite number of Pythagorean triples, like proving that 2 + 2 = 4, involves assuming that arbitrary definitions and axioms are true.

Pop Quiz: What arbitrary axioms and definitions appear in the proof in the OP?
In any case, the infinitude of such triples was first proved by Euclid...
I've long wondered how it can be proved that there's an infinite number of Pythagorean triples. I just happened to stumble on my proof when I noticed in my spreadsheet calculations that doubling the three numbers in a Pythagorean triple results in a new Pythagorean triple. I then tried some formulas based on nonnegative integers that result in Pythagorean triples, and then I saw how a different Pythagorean triple results from each nonnegative integer.
...and the proof is trivial.
Obviously Euclid and I disagree with your opinion here.

Well, the net abounds with proofs of Pythagorean Triples and infinite sets. I'd say it is common.

Today it is a common algebraic process. It looks like a form of proof by induction.

What grade level is mathematical induction?
Usually in grade 11, students are taught to prove algebraic relationships such as equations, inequalities and divisibility properties by mathematical induction. Proof by mathematical induction is a method to prove statements that are true for every natural number.

I remember it from high school math.

In the day the use of zero in counting and arithmetic was a major inbention, attrinued to the Arabs.

Well, the net abounds with proofs of Pythagorean Triples and infinite sets. I'd say it is common.
Yes. It's common for mathematicians to prove theorems online.
Today it is a common algebraic process. It looks like a form of proof by induction.
There may be a superficial resemblance to mathematical induction in my proof, but no, I didn't use mathematical induction although I might have. If I had used mathematical induction, then I would have posted a "base step" in which I demonstrate the truth of the theorem when natural number n = 0. The next step would have been the "induction step" where it is necessary that if I assume that the theorem is true for any natural number n, then it must be true for n + 1. I didn't need to do that because I was able to show directly that the triples 2^n ∙ 3, 2^n ∙ 4, and 2^n ∙ 5 are Pythagorean triples for all values of n.
What grade level is mathematical induction?
Usually in grade 11, students are taught to prove algebraic relationships such as equations, inequalities and divisibility properties by mathematical induction. Proof by mathematical induction is a method to prove statements that are true for every natural number.

I remember it from high school math.
Actually mathematicians at all educational levels study Pythagorean triples. In fact, the material I'm studying now is pre-graduate-level mathematics. So college students preparing for graduate school would study the kind of material I posted in the OP.
In the day the use of zero in counting and arithmetic was a major inbention, attrinued to the Arabs.
OK, but what does that have to do with what we're discussing?

A triangle with sides of 3 yards, 4 yards and 5 yards is a Pythagorean right triangle. How about a triangle of 9 feet, 12 feet and 15 feet? Of course it's a Pythagorean right triangle: It's the very same triangle. There are 12 inches to a foot, so (108, 144, 180) is also Pythagorean.
And so is (108 trillion, 144 trillion, 180 trillion).

When the sides have no common factor the Pythagorean triplet is called a Primitive Pythagorean Triplet.

"The essence of mathematics lies in its freedom;" so of course there are many different styles of math papers and textbooks. One paper -- an elegant new proof of Fermat's Christmas theorem -- is famous for being a single sentence in length! But as a general rule, the statements asserted in a math paper or text will be in one of five grades of difficulty:
1. Conjecture -- thought to be true, but no proof is known.
2. Theorem or Lemma -- proven with a non-trivial proof.
3. Corollary -- a theorem easily proven with two sentences or so.
4. Remark -- a corollary so trivial that the 2-sentence proof is omitted.
5. (none) -- a remark so trivial it won't even be remarked.
I've long wondered how it can be proved that there's an infinite number of Pythagorean triples.
I wonder what grade of difficulty this would be assigned.

Less trivial is to prove the infinitude of Primitive Pythagorean Triplets.
SPOILER: This will be demonstrated in the next paragraph.

Given odd positive integers s,t, relatively prime and with t < s
(ss+tt)/2 ; (ss-tt)/2 ; st
is a primitive Pythagorean triplet. Moreover, all primitive Pythagorean triplets have this form.
If we set t=1, we get a sequence of almost isosceles right triangles:
 s,t = 3,1 5 4 3 s,t = 5,1 13 12 5 s,t = 7,1 25 24 7 s,t = 9,1 41 40 9 s,t = 99,1 4901 4900 99 s,t = 999,1 499001 499000 999

One of the most remarkable facts about Pythagorean triplets is that (18541, 13500, 12709) turned up (along with other Pythagorean triplets) on a clay inscription from early Babylon, inscribed almost 13 centuries before Pythagoras! This is a primitive triplet, generated by s=179 ; t=71. How in heaven's name did they find it? Quite possibly these Babylonians knew the generating formula just given.

The n=4 case of Fermat's Last Theorem can be proven (though with considerable difficulty) by exploiting simple theorems about Pythagorean triplets.

I've never proved any non-trivial theorems about Pythagorean triplets, but they do come in handy in the construction of orthogonal integer-valued transforms. For example
( 13 0 0 // 0 12 5 // 0 -5 12 ) ÷ 13
is an orthogonal matrix.

I've long wondered how it can be proved that there's an infinite number of Pythagorean triples.
I wonder what grade of difficulty this would be assigned.
If you think this material is easy, then you've probably never studied it. I really don't care what grades of difficulty are assigned to mathematical disciplines. What matters is that the work helps to advance knowledge. As a semi-retired math educator, I try to encourage my students to work and study diligently. To mock their accomplishments as trivial is not only ignorant but serves to contribute to math illiteracy. I keep hearing people complaining about the horrors of learning math. It's no mystery that they feel that way.

It is relatively easy, if you have been around and used math.

How about Rolle's Theorem and the Mean Value Theorem, I remember them from Calculus 101. Far more important than Pythagorean triples.

Or the poof of the La Place Transform pair, wide reaching importance. Ditto with Fourier Transform pairs.

Important math proofs are widely available on the net. There is nothing mysterious to math.

I read books on neurosurgery for 10 years, therefore I am a neurosurgeon?

Here's a quick puzzle for budding mathematicians. No Fair Googling!

What percentage of the earth's surface is north of 30-degree North latitude? (Assume spherical Earth.)

Here's a quick puzzle for budding mathematicians. No Fair Googling!
Your puzzle here is an application that requires information that's not included in pure math. To prohibit Googling rules out anybody solving this problem who doesn't know the earth's dimensions or how latitude is defined.
What percentage of the earth's surface is north of 30-degree North latitude? (Assume spherical Earth.)
I've decided to post a pure-math problem that involves the mathematics that can be used to solve the kind of problem you posted. Let f(x) = √(16 - x^2) be a function in R2. If the curve of f(x) is rotated 360 degrees about the x-axis, then the result is a "surface of rotation" in the shape of a sphere of radius 4. If we want to compute the surface area of this sphere that is beyond a line through the origin that makes an angle of 30 degrees with the y-axis, then to find the appropriate value of x to start the surface we solve √(3)x = √(16 - x^2). That value is x = 2. Now we have enough information to compute the surface of revolution using an appropriate integral. See the attached file for a screen shot of that integral. The integral = 16π which is the value of the surface area in question.

Masterful--no?

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• Surface Area Integral.png
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You are hding your lack of math knowledge and experience behind 'pure math'. Mathematical problem solving is the same process regardless of your dichotomy of pure and applied math.

Take a look at the problem I posted on Math Quiz on the science form and see what you can do. It is all basic math. Calculus and algebra. A problem you will not be able to look it up online or in a book.

I read books on neurosurgery for 10 years, therefore I am a neurosurgeon?
I hope not. Just like whenever I cross a bridge, I hope you are not the engineer who designed it. But although reading books on surgery are not sufficient to make you are surgeon, reading such books are necessary to make you a surgeon. The same goes for being a mathematician; studying books is absolutely necessary to make anybody a mathematician. That's why I study math books every day. To do so is necessary for me to be a mathematician. My demonstrating my mathematical knowledge and skills is sufficient to make me a mathematician. I keep posting it. And you reply with jealous snarkery.

Bridges no.

However in the 90s maybe still today if you flew on a Boeing jet your safety in part was due to avionics I had a hand in. Scary thought aint it? Didn't work for Boeing, an aerospace company.

Aviation electronics is an area when one has to be right. The consequences have been in the news for not getting it right. The Boeing 737 Max.

Math was a spoken language in my world. You could not communicate without it.

If you are not around math then you have no reference point and you assess yourself. No peer review.

You have never made a presentaion based on your math analysis in front of critical peers. The response on the thread to your posts might give you an idea of the environmnt I worked in..

You postd a proof of Pythagorean tripples wodely available on the net. Then pewsums somw sort of authority. Have you done anything original? What is the most difficult math analysis yiu have done?

There is a difference between doing math as an avocation and math used to accomplish something tangible where being rong has consequncces.

A ciousin of mine a mechanical engineer in the 90s worked on the cryogenic cooling system for the Brookhaven RHIC partcle collider. Magmets are super cooled. One of his responsibilities was mainlining the time temperature profile for cooling and warming up the ring. If he got it wrong the very expensive ring could be damageg by thermal expansion issues.

The stuff you post is trivial compared to rreal math analysis.

Are you familar with Rolle's Theorm and the Mean Value Theorm? How about the proof of the La Place Trasfrom? A litte moore involved than simple induction applied to the Pysgorean theorem.

Something for you to chew on. 'Pure mathematics'. With wide ranging importance in theoretical and applied math.

Some consider it rude to post a puzzle without posting solution if it remains unsolved. I see such an unsolved puzzle in this thread.
Here's a quick puzzle for budding mathematicians. No Fair Googling!
Your puzzle here is an application that requires information that's not included in pure math. To prohibit Googling rules out anybody solving this problem who doesn't know the earth's dimensions or how latitude is defined.
What percentage of the earth's surface is north of 30-degree North latitude? (Assume spherical Earth.)
25.000% (¼)

Sin(30°) = ½