5, 13 and 137 are Pythagorean Primes - Numberphile
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- čas přidán 20. 08. 2024
- Professor Laurence Eaves on Pythagorean Primes - and why 5, 13 and 137 are three of his favourites!?
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When professor Eaves says "pythagorean prime", it kinda sounds like the name of some powerful magical creature, because of his voice.
Behold my creation! Pythagorean Prime!
Some new uber badass transformer
But that's exactly what it is!
It's Optimus Prime long lost brother who was suppose to be the next prime
It’s the cultured tone of the the king’s English being spoken by a highly educated subject of the realm.
137 is my favourite number!
OrlinNorris 137 is the 33rd prime
Also mine 😉
Same
@1729 math_blog the fractal universe
Wolfgang Pauli who is obsessed with alpha 1/137, he died in room 137. 🔺
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) ;)
I wonder the probability that a prime holds both characteristics (a "pythagorean prime and a prime hypotenuse if a pythagorean triple)
p = n² + m² /\ p² = j²+ k² :p is prime and n,m,j,k are integers
p² = n⁴ + 2n²m² + m⁴ = j² + k²
2 equations, 5 unknowns so it's a 3D subset in 5D space (that's not proper language, but a 5D subset with is equal to its projection onto a eD space us wordy)
Huh interesting
That's not a coincidence: it will always work :-)
Suppose p is a Pythagorean prime, and m^2 + n^2 = p, then you can construct the following Pythagorean triple: (m^2 - n^2)^2 + (2mn)^2 = p^2
one can develop these numbers by the fibonacci sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, .........................
if we take four consecutive fibonacci numbers say : 1, 2, 3, 5
multiply the two outer numbers ( 1 * 5 = 5 )
double the product of the two inner numbers ( 2 * 2 * 3 = 12 )
and can figure out the third number by 5^2 + 12^2 = 13^2
this can be done for any fibonacci numbers.
Furthermore we can find the area of the triangle generated by multiplying all four fibonacci numbers together : 1 * 2 * 3 * 5 = 30
and the perimeter is double the product of the last two fibonacci numbers : 2 * 3 * 5 = 30
Never knew that. Lovely! Don't know what you think of numerology, but "The Mystery of Numbers" gives the number 2701 (using the model of Hebrew gematria on the 26 (= YudHeVavHe) letters of the English alphabet). 2701 is the Hebrew gematria of the first verse in Torah. 2701 = 37*73, mirrow primes with mirror indices. "The Numbers Three and Seven're considered Perfect in Qabala" also gives 2701.
+travisbaskerfield
Numerology is not my thing but here's a little fun fact:
37 is number 12 on the list of primes while
73 is number 21.
Do I hear Twilight Zone music in the background?
Michael Empeigne nice property this can be seen from the identity (m2-n2)2+(2mn)2=(m2 +n2) 2 where m2 is m squared and m and n are coprime. Also this works with any sequence formed by adding the two previous terms so it's not unique to the fibonacci numbers.
I'm a big fan of Prof. Eaves maths/physics chats. 1/137 was a classic but my favourite is the Planks constant and Dirac's large number hypothesis vid. plus extra footage. It's great to see him back in action again.))
indeed.. glad you liked it!
Sure did
Yeah
I love how all of a sudden at 1:11 he pulls out a right-angle triangle out of nowhere
Also, how many pythagorean primes are Fermat primes? If a pyth. prime is 4n + 1 and a Fermat prime is 2^n + 1 then 2^n = 4n so the only solution is if n = 4 which shows that 2^4 = 4^2 which is true. Something special about 17 !!!
2 is the prime constructed by 1^2 + 1^2 unless I'm missing something. I refer to the list @ 3:11
It was just something I noticed; It's a wonderful video
cool, glad you like them... hope you're checking some of the other channels!?
when i first saw this video, i assumed they were called Pythagorean primes because they were part of Pythagorean triples (3,4,5; 5,12,13; 88,105,137; etc.). its weird how it works for the same numbers (17 as well)
I've posted Professor Eaves' old sixtysymbols video on 137 (the fine structure constant) as a video response and it the video description!
Hi
137
HAPPY BIRTHDAY, MATE
This is starting to become one of my favorite channels.
You think of great questions to ask, Brady!
I love this guy's accent and voice.
Someone else who loves 137
137 is seriously my favorite number.
I love it. Did u know what the mass number of barium is? U will never guess. It is 137
137 is the number of degrees turned when using the golden ratio for dropping seeds around a plant.
137, 173, 317 are all primes. The next three permutations in ascending order, 371, 713, 731, and the mirrors of the previous, are composites. Unique?
Does 1307 count as 137? My apartment unit is 1307 😃
Fun stuff! My only disappointment was failure to point out that, while for each Pyth. prime, p, √p is the hypotenuse of a rt. triangle with integer legs (a, b), so is p itself! (with bigger legs, of course)
And you can get the legs for the bigger triangle, by squaring the corresponding complex integer with the smaller legs:
(a + bi)² = a²-b² + 2abi
So for p = 137 = 11² + 4², you have 11² - 4² = 105; 2·11·4 = 88, so:
137² = 105² + 88²
The point is that when you plot these in the complex plane, the length is the Pythagorean sum of the x (real) and y (imaginary) components:
z = x + iy ; |z| = √(x² + y²)
All this really says is that if you draw the corresponding right triangle, with a along x, and b along y, then c, the hypotenuse satisfies the familiar Pythagorean Theorem:
a² + b² = c²
Now when you square a complex number, its length also gets squared. Which means that the squares of the new components (A=a²-b² and B=2ab), also satisfy the Pythagorean Theorem, with:
A² + B² = C² = (c²)²
(a²-b²)² + (2ab)² = (a² + b²)²
So in my example,
a=11, b=4, c=√137, c²=137
A = a²-b² = 105 , B = 2ab = 88 , C = c² = 137
C² = 137² = 105² + 88²
Yes, it's one of those totally marvelous techniques that makes me wish I'd thought of it!
done that one already!
Really nice to see Prof Eaves again!
137, of course!
The example he gave us, which was A = 3.2 inches and B = 5.75 inches, then C is about 6.5804635095105572661826405151142 inches.
My daughters birthday 17th May so my favourite is 175 including Matrix . 71, 157 half of Pi, 571 and 751. Cheers. All Primes
a quick way to find them all is my website - bradyharan com
I wonder how many Pythagorean primes are also hypotenuse lengths for Pythagorean triples. Both 5 and 13 are. 5 = 2^2 + 1^2, but also 5^2 = 4^2 + 3^2. And 13 = 3^2 + 2^2, but also 13^2 = 12^2 + 5^2. And 17^2 = 15^2 + 8^2. And 29 = 5^2 + 2^2, but also 29^2 = 20^2 + 21^2. etc.
And...
wait for it...
137^2 = 105^2 + 88^2
but 2=1^2+1^2 and 2 is prime, while there exist no 2 natural numbers a and b such that 2^2=a^2+b^2 :(
All of them are, in fact! It's a known fact that all Pythagorean triples take the form (2kmn, k(m^2 - n^2), k(m^2 + n^2)), with k, m, and n integers with either m or n even, and m > n. I could prove that, but the margins of this comment are too small (in all serious, it's not that hard of a proof). So all Pythagorean triples look like that. In all of them, k(m^2 + n^2) will be the hypotenuse. So you can see that if k = 1, then the hypotenuse can always be represented as m^2 + n^2 for integers m and n. So if m^2 + n^2 is a prime, then it's also the hypotenuse of some right triangle. :)
+Kieran Kaempen also, m != n.
It is the SQUARE of the hypotenuse of an integer right triangle that might be a Pythagorean prime - not the square root of the hypotenuse.
Jim Harmon but a square number can never be prime : p so that is a really silly definition
I will parenthesize my statement to make it more obvious what I meant:
It is the SQUARE of (the hypotenuse of an integer right triangle that might be a Pythagorean prime). That is, the value of the hypotenuse being the prime, not its square.
Some salient facts, not all of which are brought out in the video:
1. Every prime is either 2, or is congruent to either ±1 (mod 4).
2. Every prime that is 2 or == +1 (mod 4), and no other prime, is expressible as a sum of two squares.
3. Every number, n, that is expressible as a sum of two squares, and whose sqrt, √n, can thus be the hypotenuse of a right triangle with integer legs (a,b, so that a²+b²=n), can also itself be the hypotenuse of an integer right triangle, by virtue of squaring the complex integer a+bi, and using the squared length of that complex integer:
(a+bi)² = a²-b² + 2abi ;
(a²-b²)² + (2ab)² = (a²+b²)² = n²
The value of the length of the hypotenuse squared which would be the Pythagorean prime. In His example, for instance, a right triangle with sides a=11 and b=4 would have a hypotenuse of length 137^(1/2). We can show this by applying the Pythagorean Theorem: a^2 + b^2 = c^2, for a=11 and b=4 would be (11^2)+(4^2) = c^2 = 137 (the prime number.) To get the length of the hypotenuse, we would need to take the square root of our Pythagorean prime, which would be roughly 11.7047.
Parker square of the hypothenuse
137 is one of my favorite numbers because it's Porygon's number, which is my favorite pokemon. Okay now I feel really nerdy... =/
Finally one on numberphile who writes the "7" correctly. It's supposed to be stroked. (And I could tell you why...)
Tell me
Because in the original version of how to write the set of symbols we still use to express numbers today, each symbol contained the same number of angles within itself as the number the symbol was trying to express. (Just the single digits 0-9 of course.)
0 has zero angles.
1 as it typically appears in type has one angle.
2 is written like a ‘z’, and thus contains 2 angles.
Make the curves of 3 into points instead. Three angles.
Look it up. Some numbers are slightly trickier to explain in words, but they are completely recognizable when you see a picture.
You’re welcome ;)
what are the other channels besides sixty symbols...i'm addicted to these videos.
I was wondering whether the connection between 137 and the fine structure constant would be mentioned.
I'm a Pythagorean prime baby!!!
I too love the number 137, it has been my favourite for a long time, just randomly (kind of). Over time I keep finding out cool things about it :D
Blaise Pascal approves. (73^n)(137^n).
Leonardo Bigollo Pisano approves. (11^n)(101^n)(73^n)(137^n).
"Come find me at the Cliff if you want to scale the mountain." ~Meru
Note: if m^2+n^2 = c, for integers m and n, then there exists integers a and b such that a^2+b^2 = c^2. (By Euclid's Formula). I'm surprised that wasn't in there....
My birthday is the 13th may too :D It's kind of satisfying.
+BibiCookiecat Same!
5:00 Its *square* is a Pythagorean prime.
I'm probably the millionth person to post this.
5.13 is also my brothers birthday and 513 is one of those numbers I see everywhere, this video is just another
Happy Birthday :)
It just so happens that 5 and 13 are also the hypotenuses of right-angled triangles when the legs are integers. Because sqrt(3^2+4^2)=5 and sqrt(5^2+12^2).
ya, this is what i meant
prefix this with the definition that a pythagorean prime (PP) is a prime of form 4n+1
this is equivalent to saying the definition that PP is the sum of two squares and prime
and that is equivalent to the definition that a PP is a prime hypotenuse of a pythagorean triple
What about two? Two is prime.
1²+1²=2
Is it not also Pythagorean, despite not following the 4n+1 property?
The thing is, they don't count as prime.
John Galdino 2 is prime. Just the weird one that doesn't follow the others. 1 however isn't considered prime.
Mageknight, you're right.
c = sqr(2) in the special case where a = b = 1, so if they say all pythagorean primes come in the form 4n+1, then I guess they add the condition that
"a is not equal to b", or in other words: the triangle must be a half rectangle, except for the special case where the rectangle is a square.
does a triangle only have two sides?
We are in Earth dimension C137
I remember that I got 137 marks (out of 200) in physics exam 2 years ago. XD
Interestingly, 7 x 3 = 21 and 73 is the 21st prime.
My youtube account used to have the number 137 in it, because I thought it was an awesome number. Now I have another reason to like it.
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!
The month I was born on was the 6th and the 28th day. its a perfect number day, and its the only one in a year!
What about 6/6?
in fact they are equivalent. if p is one of these pythagorean primes as described above, then p=a^2+b^2, which can be factored into a product of complex conjugates
p=(a+ib)(a-ib)
then p^2=(a+ib)^2*(a-ib)^2=(a^2-b^2+2abi)(a^2-b^2-2abi)
this again is a product of complex conjugates, so
p^2=(a^2-b^2)^2 + (2ab)^2
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.
yeah it was my very clumsy question that caused the problem... but I think you know what was meant!
"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.
cuz the diagonal can be calculated as a^2+b^2=c^2 and then take the square rot of c^2 and u get the diagonal. and he used the same method to get 5, 1^2+2^2=c^2, c^2=5
Prime numbers (especially large ones) are important in cryptography. Pythagorean primes can be generated by the constraints a^2+b^c=c and 4n+1=c; having a nice way to generate prime numbers, even a subset of them, is useful.
137!!!
Thats my favorite number =)
same =D
1337 xd
137 is the 15th Pythagorean prime.
11 squared is 121
4 squared is 16
121 + 16 = 137
11 + 4 = 15
113 is the 14th Pythagorean prime.
10 squared is 100
100 + 13 = 113
11 + 3 = 14
109 is the 13th Pythagorean prime.
10 squared is 100.
3 squared is 9
10 + 3 = 13
has anyone noticed that the fine structural constant is almost the same as the half the decimal in pi. that 137 has a repeating decimal of 8. and mass as a wave repeats after 8 turns.
137 is an inverse of a fine structure constant
I like 233 because the roots of the perfect squares which add up to it (8, 13) are the parts of the wedding anniversary of a cousin of mine.
And also all these three numbers are in the Fibonacci sequence!
Aren't these all just primes that are one more than a multiple of four?
Neel Modi Yes, that's what he showed with the formula 4n+1
***** oh yeah oops
You now have 5 likes, which is one more than a multiple of 4 :)
It seems like these are all primes that are the sum of two squares as well.
5 = 1² + 2²
13 = 2² + 3²
137 = 4² + 11²
Notice that each pair of the bases of the squares are all co-prime.
You now have 9 likes, which is one more than a multiple of 4 :)
do you know that 5/13 will be lucky in 2017 because :5/13/17
2017 is also prime, but I don't know if it's a Pythagorean Prime.
Jeff O Yes, it is.
Jeff O 44^2+9^2
The fact that it's prime and congruent to +1 mod 4, guarantees that it is expressible as a sum of two squares. Also note that
1855² + 792² = 2017²
[This is always possible for Pythagorean primes.]
Just to say that every positive integer k of the form 4n+1 can be expressed as the sum of two squares provided that
a) the prime factorisation of k consists only of primes of the form 4n+1 or
b) if the prime factorisation of k consists of any primes of the form 4n+3, then each such prime factor must be of a power having an even exponent. p = 4n+3 gives p² = 4(4n² + 6n + 2) + 1
which is of the form 4n+1
Given the possible remainders on division, the integer z = x² + y² can be of the form
4n, 4n+1 or 4n+2
but it is impossible to arrive at c = 4n + 3
This explains why 21 = 3 *7 cannot be expressed as the sum of two squares; each prime factor has odd exponent 1. However, if you consider 45 = 3² * 5 for example then
45 = 3² (1² + 2²) = (3² + 6²) = 45
Assuming that an integer can be expressed as the sum of two squares, the number of ways it can be so expressed, depends on the number of prime factors (up to exponent) which are of the form 4n+1. Any prime factors p^2r with p of the form 4n+3 are discounted in the calculation since their product is simply factored into each particular expression involving the former primes, with primes of the form 4n+1 being expressed as the sum of two squares in only one way. For instance, all positive integers k with prime factorisation
k = pq (where each of p and q are congruent to 1 modulo 4) will have two ways of being expressed as the sum of two squares. For example
65 = 1² + 8² = 4² + 7²
This comes from the fact that if P and Q are primes (of the form 4n+1, naturally), then since we can write
P = a² + b² and Q = c² + d²
PQ = (a² + b²)(c² + d²) = (ac + bd)² + (ad - bc)² - the sum of two squares
But we could also have
PQ = (a² + b²)(d² + c²) = (ad + bc)² + (ac - bd)²
which gives the two expressions.
so
5*13 = (1² + 2²)(2² + 3²)
= (1*2 + 2*3)² + (1*3 - 2*2)²
= 8² + (-1)²
= 8² + 1²
and
5*13 = (1² + 2²)(3² + 2²)
= (1*3 + 2*2)² + (1*2 - 2*3)²
= 7² + (-4)²
= 7² + 4²
(I think I got a little carried away here......)
I'm assuming you're referring to the sums thatyield a pythagorean prime. (Such as 1^2 +4^2= 17. In this example neither integers are primes. A Pythagorean prime is the sum of two *integers* squared, not necessarily two primes that are squared.
Born in May 13 and 137 years old :)
Simoss13 if he ever turns 137 that will be his favorite birthday
I suddenly realized that 149 (7^2+10^2) and 181 9^2+10^2) are Pythagorean primes. Judging from that list, they seem quite common.
quote:"a pythagorean prime (PP) is a prime of form 4n+1" i found this definition a number of different places on the web
137 Its something special about that number i see it occur many times a day for example 15-20 times while others only does like 1-5 for me
Yes it would... Side lengths 1, 1, and √2. It's an isosceles right triangle.
Pythagorean primes are special because they fit both the pattern of Pythagorean numbers and primes. That's it.
Mathematicians love this kind of thing. I mean they make such a deal over Mersen primes and prime numbers in general for example. It's really quite arbitrary.
Not to say I don't love mathematics, because I do, and it is cool to find patterns or particularly rare occurrences, but a lot of it seems arbitrary if you lack higher understanding like I, or most people do.
oh I forgot to do that trick! :)
I think this is the first time anyone has ever messed up on one of those brown sheets of papers that are in Brady's videos
Professor Laurence has really pretty numbers!
Neat. Happy birthday!
So THATS why our Rick and Morty reside in universe c137! They said it was arbitrary!
... 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.
Hey Numberphile, I remember there were combinations of A and B where B=A+1 (1 different to A)
Though I forgot how to find them.
The easiest example of course is if A=3 so B=3+1=4 and so C=5 :D
Here's something else about the numbers 5 and 13:
5 is the number of platonic solids &
13 is the number of Archimedean solids! Both very closely related.
@numberphile You should do a video on the Dyson Number!
He has very neat writing.
hi, my other channels include sixtysymbols, periodicvideos, nottinghamscience, deepskyvideos, foodskey, backstagescience, etc...
Oh yeah.. A bit more to explain it.. 2 would look like Z, 3 would look sort of like like epsilon, and then you get the point.. 8 is just two squares on top of each other..
I was doing my homework on page 137 today.
I LOVE your videos and I have watched them all, but the sound of the markers on the paper makes me shiver to the bone!! Please use a white board, you'll make me one happy viewer.
I'm glad they didn't listen to you xD
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
64 and 137 are my two favourite numbers :) in 137, 13 is prime, 37 is prime, and 1+3+7 is prime, 3 is prime and 7 is prime, and 137 itself is prime. One of the best numbers out there imo :) And I love 64 because 2^6 is 64, 4^3 is 64, 8^2 is 64 and 16^1.5=64 which I think is awesome because I love powers of 2, and I love binary. When people ask me what my favourite numbers are, they thing that they're really strange, lol.
My birthday is 5/29 and my brother's is 5/17! We are pythagorean prime brothers!
Welsh understatement!
My results are curiously exacts and no many computes are required.Consider each number separetly.Parity is primordial.For ex. the parity of number of digits,the parity of last digits, and so on.The sum of them and parity.The digit currently test is prime number? It,s works likes a zero.
137 is also the rounded down golden ratio (in degrees) :)
Theres one thing I do not understand. Why does he take the square value as the "Pythagorean Prime"? Because that is not the actual value of the hypothenusis, the value is actually the square root of those numbers. And in all the cases he illustrated, the actual value of the hypothenusis is an irrational number, which isn't that special. Why doesn't he take the "proper" approach? 5^2 = 4^2 + 3^2 for instance?
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.
If a^2 + b^2 = c, and a and b are integers, and c is prime, then c is called a "Pythagorean prime" because it matches the form of the Pythagorean Theorem.
I was born on 05-11-1997. and the 5th,9th, and 1997th digit of pi is 5.
Yea, was gonna correct myself but whatever, thats why it isnt a rational. yea and Phi has also the property of Phi²=Phi +1, and Phi³= (Phi +1)/(Phi -1). And my favourite, but im not sure : Phi= -cos(6x6x6) (in degrees, not radians)
It's based on the number of angles really. 0 has 0 angles, 1 has 1, 2 has 2, 3 has 3, so on and so on. And to get a 7 with 7 angles you need the stroke. Of course, as we write them now, you most often don't have any angles at all apart from 2, 4, 5 and 7, the rest is just rounded. And oh, the 4 as you see it here, in this text, is wrong too because it has 6 angles, not 4. But it's not a big deal I guess.. I just really hate it when people write the 7 without the stroke : P
Thank you
I always wanted to know if you can make a right angled triangle with these primes making up the edges
game of numbers: 137(n=1,3,7) 2^n+2+n 2^1+2+1=5 2^3+2+3=13 2^7+2+7=137
wouldn't 2 be one too?
+Holly Hills I guess so.
Holly Hills When he talks about Pythagorean primes, he's talking about the integers that can equal C. Assuming A = 1 and B = 1, A² + B² = 2. If C² = 2, C = √2, which is neither a prime nor an integer.
Lane Duncan No, check the video again. It says 5 = 1^2+2^2, but not 5^2 = 3^2+4^2
Yes, two is the only prime with that property who is not of the form 4n + 1
Porygon 2's number is 233 which is another Pythagorean Prime number.
No, there are not. See Fermat's theorem on the sum of two squares. Many of the proofs (e.g. Dedekind's proof) implies that if the sum of two squares is prime, then those two squares are unique.
wikipedia has a nice treatment of Pythagorean triples