5, 13 and 137 are Pythagorean Primes - Numberphile

Professor Laurence Eaves on Pythagorean Primes - and why 5, 13 and 137 are three of his favourites!?
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NUMBERPHILE

Videos by Brady Haran

[georgemissailidis]

I love how all of a sudden at 1:11 he pulls out a right-angle triangle out of nowhere

[aesdeef]

Before I saw the video I thought you'll be talking about Pythagorean triples where the lenght of the hypotenuse happens to be a prime number. It actually works for all the three numbers mentioned in the title: (3, 4, 5), (5, 12, 13), (88, 105, 137) ;)

[numberphile]

I've posted Professor Eaves' old sixtysymbols video on 137 (the fine structure constant) as a video response and it the video description!

[RoaringTRex]

You think of great questions to ask, Brady!

[aletoledo]

This is starting to become one of my favorite channels.

[stygn]

Finally one on numberphile who writes the "7" correctly. It's supposed to be stroked. (And I could tell you why...)

[lbaconstrips]

I love the # 137 because a used to go on route 137 on vacation. This was when i was a little kid. Now seeing this math is really cool!

[ISmokePopRocks]

5.13 is also my brothers birthday and 513 is one of those numbers I see everywhere, this video is just another

[numberphile]

yeah it was my very clumsy question that caused the problem... but I think you know what was meant!

[AtheistCitizen]

k >0 forces the lengths A, B, C of the corresponding rt triangle to be positive. If you want to "play" with sign and do not associate PT with a triangle then +/- any A, B, or C but this will not add new Natural number solutions, hence no new triangles.

PP with a=b solution is unique at PP=2. Other a=b solutions? Wd imply prime divisible by 2 [so no others]. Silly case to consider. So a or b, one must be larger. WLOG let a>b keeps all A, B, C positive so the solution corresponds to a right triangle.

[michaelbayer]

... u can go also to state delaware - the First state of Amerika - the mountain so high - 137 the lowest high mountain Position in Delaware called - Ebright Azimuth - its like Gate to the STARS.

[stumbling]

"Also it's square root is not an integer", prime numbers don't have integral square roots because they are non-divisible (hence the fact they are primes). Ten doesn't have an integral square root, but the reason it isn't a prime is that it's the product of two primes (2 and 5). Only numbers that can't be reduced to smaller "parts" are primes, they are the building blocks of all numbers. The number one doesn't count in this sense because dividing by one doesn't reduce the number to smaller parts.

[hooliaw]

nice video Brady! but i thought 1 is not a prime number (as said in one of ur video)?

[theGraphicAutist]

So THATS why our Rick and Morty reside in universe c137! They said it was arbitrary!

[stumbling]

I don't know how I missed the middle part of your comment but I did. I still don't see where he says 10 is a prime though.

[APMathNerd]

Yes it would... Side lengths 1, 1, and √2. It's an isosceles right triangle.

[neelmodi]

Aren't these all just primes that are one more than a multiple of four?

[TheMdc]

right which would make the actual length of the side some odd decimal number, but the square of that number is 17 so the OP is correct, they misspoke

[JonathanTot]

quote:"a pythagorean prime (PP) is a prime of form 4n+1" i found this definition a number of different places on the web

[bxdanny]

While the triangles he shows have only the square roots of these "pythagorean primes" for their hypotenuses, they seem to be numbers that can themselves be the hypotenuse of a right triangle with integer sides (5 for 3/4/5, 13 for 5/12/13, etc.) I'm wondering if this is always true.