#prime numbers

[karpholmes]

(Mathematician here)

Quite possibly the funniest targeted ad I’ve ever gotten

[noelements-setempty]

Stop asking me for my phone number you bitch ass websites. The only 'two factor' things that interest me are the beautiful prime numbers 💙

[the-psudo]

Each colored wave represents a different prime number, from 2 up to 101.

[positivelyprime]

Eisenstein primes (and integers)

You've heard of integer primes, but have you heard of different kind of primes? One of the most beautiful pictures in mathematics I've recently seen is this:

This is a picture displaying the Eisenstein primes and the rest of the post will be me explaining where this comes from.

Firstly, to understand what Eisenstein primes are, we need to know what Eisenstein integers are! An Eisenstein integer is a complex number of the form z = a + b w where a,b are integers and w = e^(2/3 pi i). Normally a small letter omega is used instead of w. Notice that w is a third root of unity, i.e. w³ = 1.

Appending this complex w to the regular integers has some consequences. In particular, we can now view the eligible numbers on the complex plane to get this picture:

The Eisenstein integers correspond exactly to places where two dotted lines cross. As such, where normally one can view integers as laying on a 1-d line, we can now view Eisenstein integers as laying on a 2-d lattice!

Now one may start to wonder if, just like in the integers, one can have Eisenstein integers which are not divisible by other Eisenstein integers. In more mathematics: We call a number p an Eisenstein prime if p cannot be written as a*b where a,b are Eisenstein numbers which are not +-1, +-w, +-w² (the units in this system).

It turns out that some regular primes are still primes in the Eisenstein integers. For example, 2 can still not be decomposed. However, 3 can! We may write 3 = -(1+2w)(1+2w) which means that 3 is not an Eisenstein prime. One may investigate this further (using abstract algebra) to arrive at the picture at the top of this post. The 6-fold rotational symmetry is due to the Eisenstein integers having 6 units.

Of course a natural question is why take a third root of unity and not an nth root? This is a very interesting question and leads to the study of (integer rings of) cyclotomic fields! This concludes fun math fact tuesday

[incognitopolls]

We ask your questions so you don’t have to! Submit your questions to have them posted anonymously as polls.

[fionaapplerocks]

Co-incidence or ???

FTBC was released on April 17 -- 17 is a prime number The Idler Wheel was released on June 19 -- 19 is a prime number Tidal was released on July 23 -- 23 is a prime number Extraordinary Machine was released on Oct 3rd -- 3 is a prime number Fiona's Birthday is September 13 -- 13 is a prime number When The Pawn was released on November 9, which is NOT a prime number, but it was the 313th day of the year. -- 313 is a prime number

primes (1 in 3 of days in the month) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

[janmisali]

while we're yelling about primes. i have never stopped being mad about 1 not being a prime. i don't understand it and no one has explained it in a way that makes any sense.

oh I know this one!

so, historically, one was considered to be a prime number, though not universally. if wikipedia is to be believed, publications as recent as 1956 included one in lists of primes.

the problem was that there's a lot of really important theorems about primes that are only true if you ignore one. the biggest is a theorem so important that it's literally called "The Fundamental Theorem of Arithmetic", that every integer greater than one has a unique prime factorization. if one is counted as a prime number, this is no longer true (you can multiply any number by one arbitrarily many times and get the same number), so older definitions of the Fundamental Theorem of Arithmetic would refer to "prime numbers greater than one".

and this context of primes as the components of a number's prime factorization is like, The context in which mathematicians care about prime numbers. it is the reason studying prime numbers is even a thing anyone does. so, if one were considered a prime number (as it once was), the word "prime" would basically only ever be used in A) lists of prime numbers and B) the phrase "prime number greater than one".

so, in a sense, you're right, it would make more sense if one were classified as a prime number. but the purpose of the words and symbols used in math is to describe the properties that mathematical objects have, and just in terms of practicality, "an integer which isn't evenly divisible by anything other than one or itself" just isn't as useful of a definition as "an integer which is only divisible by two factors, one and itself".

[myothertardisisonthemun]

I was reading "How to invent everything", and got up to this part:

And my next thought was - doesn't that sound like a doctor who story idea? A machine that finds super large primes, ones so big people aren't likely to use them, and burns them for fuel.

Untill one day, people on that planet start having problems with their mathematics......

[needlesandnilbogs]

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

[blake447]

So here's an interesting fact I've learned while researching these things called Wolstenholme Primes. See they're related to the binomial coefficient and thus pascals triangle. We've probably all seen sierpinski's triangle in here (if not circle all the odd numbers). But here's something else interesting. If you look at the triangle, you'll see that for prime numbers, every element is divisible by that prime number. Look at 11 for example.

55, 165, 330, and 462 are all divisible by 11. This has to do with the binomial coefficient definition, in which you'll see that if unless we choose either all the elements, or none of them, the factor of our prime number is guaranteed not to cancel out from the numerator's factorial.

We can then use the fact that the sum of all the elements in the nth row of pascal's triangle adds up to 2^n. Once we remove the 1's on each end of the triangle (hence the minus 2) we know that for prime numbers p that all elements are divisible by p, their sum is as well!

Now, I don't know off the top my head if this is only true for prime numbers. It isn't immediately obvious imo. However, if this is strong enough to be an if and only if, then I think I have independently discovered perhaps the *worst* way to test for primality lmao, scaling exponentially instead of the standard logarithmically.

[positiveintegers]

619

[karpholmes]

Just realized a very obvious connection between Bernoulli numbers and primes; will be uploading soon— ofc it’s related to the Riemann zeta function— anywayz: I’ve come to ask for resources for analyzing the zeta function. I mostly use Wikipedia but it’s really difficult to understand due to it relying on pre-existing concepts— basically I’m looking for one of those, “10 hour courses,” on the zeta function or associated ideas

[art-of-mathematics]

Sierpinski-triangle and Pascal"s triangle

Pascal"s triangle

I used transparent/translucent paper to color the numbers of pascal"s triangle.

Each color represents a prime number:

Red: 2

Orange: 3

Yellow: 5

Green: 7

Blue: 11

Violet: 13

For each layer (and the according prime number) I colored the numbers of pascal"s triangle if the pascals triangle number is divisable by that prime number.

The advantage of this transparent/translucent paper I used:

You can hold multiple layers against the light (and see the overlapping colors):

2 and 3:

^For instance: if you look at the 6 in the 5th row it is a combination of red and orange - 2 and 3 - as 6 is built up by the primes 2 and 3 (2*3)

2 and 3 and 5:

3 and 5 and 7:

[Afternote: I just realized I forgot to color some numbers in the modulo 2 sheet (red). ]

[prime-factorizer]

Happy Pi Day!!

I was watching Matt Parker's old pi videos and he mentioned a method of calculating pi using the prime numbers, so I figured that'd be a fun thing to do for this blog

I'm not sure how accurate it'll end up getting, so I'm just gonna leave it running for the rest of the day and give an update whenever I call it done. So far, it's gotten 4 digits but idk if it'll be able to do more

Code and video I mentioned below the cut

Constructive critisism on the code isn't discouraged. I don't plan on making money with my programs, but it would be nice to be able to get better at it

He talks about the method I'm doing at 1:38

[positivelyprime]

After a week of checking and some 6 years of waiting, we have finally found a new largest prime!!! This modest number is given by

p = 2^{136279841} - 1

and has more than 40 million digits! This was, of course, found by GIMPS.

Source:

[msexcelfractal]

Multiply the first ten primes:

2*3*5*7*11*13*17*19*23*29 = 6469693230.

Notice the pattern?

6469693230 = 2003001*323*10.

Since 323 = 17*19, that means:

3*7*11*13*23*29 = 2003001.

Well, that number factors like 2x² + 3x +1, doesn't it?

7*11*13 = 1001 and 3*23*29 = 2001.

And 1001 factors like x³ + 1:

10³ + 1 = (10 + 1)(100 - 10 + 1) = (11)(91) = 11*13*7.