Paterson Primes (with 3Blue1Brown) - Numberphile

Grant Sanderson (from 3Blue1Brown) discusses the briefly brilliant discovery of Paterson Primes. More links & stuff in full description below ↓↓↓

With thanks to Michael Colognori for computing the big prime list.

Correction at 1:20 - 5 (not 17) is 11 in base 4, - still prime of course.

And with thanks to Patrick Paterson - of course.

NUMBERPHILE

Videos by Brady Haran

Special thanks to our friend Jeff for the accommodation and filming space.

[eyflfla]

Patrick Paterson and his patented primes were a Parker precursor. He gave it a go, and got pretty close.

[maltezachariassen]

I will never not be amazed by Grant's seemingly natural understanding of complex patterns in mathematics. And it helps that he is able to calmly and precisely explain it.

[steveb]

Whenever I see the word "prime" or the name "3blue1brown" in a Numberphile video, I feel the urge to watch immediately, so I dropped everything for this one. The traffic behind me can wait until I'm done.

[edwardberryman]

I love that as an aside Grant explained the rule for finding if a number is divisible by 3 or 9. I've been using that fact for almost two decades and had never thought to ask why it was true.

[andrewharrison]

1) If you don't generate the hypothesis then you have no chance of getting a theorem.
2) When you test a hypothesis you will get a deeper understanding. Even while disproving it.
3) and it's fun.
Thumbs up to all concerned.

[Hepad_]

Makes me think of my 10 years old self, so proud of discovering that the hypothenuse of a 3 and 4 units sided right triangle is 5, and that it works for 6, 8 and 10 too.

[vigilantcosmicpenguin]

I'm jealous of Grant for having had friends like that in high school, who could just talk about nerdy math stuff. That's the coolest kind of kid.

[Astromath]

Some questions that come to mind:
- Are there infinitely many "Paterson primes"? (I do think so but can't think of a straightforward way of proving it rn)

- How exactly does the ratio between "Paterson primes" and "non-Paterson primes" behave for larger and larger numbers?

- Is there a longest consecutive run of "Paterson primes"? So, could it theoretically be all "Paterson primes" after a certain number? If so, from what number on is that?
If not (which is probably more likely), what's the longest consecutive run of "Paterson primes" we know of?

[johnchessant]

None of the Paterson composite numbers shown in the video are divisible by 11. For those wondering, the first one is 9, 011 -> 2, 030, 303 = 11 × 379 × 487.

[jamesimmo]

That ending (the first 1, 000 primes checked) was therapeutic (although it almost felt like Patrick’s obituary)

[mrmorganmusic]

This is great! I love seeing my favorite YouTubers entering each other's worlds. I did notice a typo (others probably did too): at 1:17, the graphic indicates that we are writing 17 in base 4, but the prime in question, as Grant just stated, was 5 (11 in base 4).

[the_box]

So much editing for part 3. I bet it's going to be amazing!

[exoplanet]

5:00 I've used the "add the digits" trick to check for divisibility by 3 for years...but never knew why it worked.

[konstantinrebrov]

I really enjoy the work of 3Blue1Brown. He has a way of explaining things that just intuitively makes sense.

[fuuryuuSKK]

Oh hey, look at us breaking into the numberphile "prerelease vault"

[themathhatter]

After some thought, I've come up with an extension to Paterson Primes.

Consider a set of primes {p1, p2...pn} and a small base A. To find a larger base B such that, when you take a prime in base A and interpret it in base B, will not divide any of the primes in the set, B must be subject to the following conditions: if a prime from the set p is larger than A, B=k*p for some natural number k, or in general, A=B (mod p).

Let's do a small example. For the set {2, 3, 5, 7}, and starting base 4, B must be a multiple of 2, one more than a multiple of three (which combine to require B is congruent to four mod six), either have a residue of four mod five or be a multiple of five, and either have a residue of four mod seven or be a multiple of seven. The smallest B which satisfies these conditions is 70. Thus, if you write the primes in base four and interpret them as base 70, you can be ensured that the resulting numbers will not be divisible by 2, 3, 5, or 7, which is neat, but far less elegant than P. Paterson's original result.

[cvoisineaddis]

Grant's eloquence and conveyance of mathematical principles is near unmatched.

[Chalisque]

The add-the-digits test for divisibility by 3 was my first experience of discovering a proof of a result. It was a bonus exercise my older sister had been set in school. Addictive experience.

[AngryArmadillo]

I feel like we have definitely observed an increase in Grady’s mathematical abilities/confidence over the years of him conducting all these wonderful interviews. Love to see it!

[rockallmusic]

The 3Blue1Brown channel dropped a new video only a couple of hours ago and now we get THIS TOO today??? Christmas came early!