Near repdigit primes on Numberphile

[lavalamp]

Thought some people on here may find this interesting, popularising of interesting primes.
https://www.youtube.com/watch?v=HPfAnX5blO0

[Batalov]

It is nice for popularising, sure, but they still ought not to make up "new names" when established terms exist.
It is great to explain everything as well as they do (this episode is no outlier, they always very well done on the explanation and element-of-surprise level), but without reference to the body of existing work (perhaps at the very end) it comes across as crankish. Sorry, guys, if you will be reading this. Nice, yes, ... but crankish.

They should invest some time in research before filming. Filming takes quite some effort and it is, in this case, done professionally, but what does a good production company have besides actors (be that in this case 'actor/author/producer/director') and an operator, a lighting and sound crew? There is always research.

"Palindromic near-repdigit primes". LMGTFY!
UTM pages, as well as worldofnumbers.com, recmath and OEIS immediately show up in this search. No effort needed.

[Batalov]

* OEIS A265383 (pending approval)
* Kamada's NRR (these are "9v89w" in his notation, i.e. v w; the 9w89w are palindromes, 10²ⁿ⁺¹-10ⁿ-1, studied much deeper by Darren Bedwell, with the largest known prime 10^134809-10^67404-1).
* UTM NRR primes
* H. C. Williams, "Some primes with interesting digit patterns," Math. Comp., 32 (1978) 1306--1310. Corrigendum in 39 (1982), 759. MR 58:484

[CRGreathouse]

Originally Posted by Batalov

* OEIS A265383 (pending approval)

Approved.

[Mini-Geek]

I wonder if the database he referred to in the video was factordb.com?

Originally Posted by A265383

1, 6, 9, 154, 253, 1114, 1390, 2618, 5611
...
15286 is a member of the sequence, due to Kazuyoshi Asao (Feb 11, 2002), but its position in sequence is currently unverified.

I've started a small (1 core) search with PFGW running from n=1 to 20k. Currently at n=4500, no surprises. I'm not sure if there has been a more thorough search or if anyone is running a similar search now.

Originally Posted by Batalov

It is nice for popularising, sure, but they still ought not to make up "new names" when established terms exist...

I'm a Numberphile fan, but I still agree with you. I guess halfway-good math that gets people interested is better than no math at all.

Originally Posted by Batalov

(be that in this case 'actor/author/producer/director')

In this video, Simon Pampena is the "actor" and Brady Haran is the "everything else" (as I understand it). Brady makes a lot of videos on math, computers, etc., but rarely stars in them.

[lavalamp]

Originally Posted by Mini-Geek

I wonder if the database he referred to in the video was factordb.com?

I had wondered the same, but it doesn't seem to be as factordb stops at 253.

[Batalov]

There is an invisible extra member in the sequence "1, 6, 9, 154, 253, 1114, 1390, 2618, 5611, 12871" (in draft).
Statistically, the next term is likely Asao's, I pre-sieved and I am scanning up to 20,000++, too.

Btw, prime UTM database remebers every little prime submitted many years ago.
Use http://primes.utm.edu/bios/page.php?id=183 and press "All of this Person's primes"...

P.S. You will find that Asao was also curious about "Cyclop"-like series 2²ⁿ-2ⁿ-1 which unlike the series in the Numberphile episode (the palindromic one, 2²ⁿ⁺¹-2ⁿ-1) is not algebraically factored. These are now, of course, also known as OEIS A098845.

[Mini-Geek]

Originally Posted by Batalov

There is an invisible extra member in the sequence "1, 6, 9, 154, 253, 1114, 1390, 2618, 5611, 12871" (in draft).
Statistically, the next term is likely Asao's, I pre-sieved and I am scanning up to 20,000++, too.

I've tested through n=13043 and found the same list as you. I'm stopping there, since you are ahead of me.

How did you pre-sieve this sequence? Is there a tool out there that can take this form, or is it easy to tweak some siever to take a general-form number, maybe? I just ran PFGW with -f so it tried factoring each number before doing its N+1 test.

[science_man_88]

Originally Posted by Mini-Geek

How did you pre-sieve this sequence? Is there a tool out there that can take this form, or is it easy to tweak some siever to take a general-form number, maybe? I just ran PFGW with -f so it tried factoring each number before doing its N+1 test.

my first thought based on the numberphile video would be the polynomial form $ x^{2n}-x^n-1 $ all numbers in the sequence for x=10,can't be divisible by 2,3,5, mod 7 it becomes 3²ⁿ-3ⁿ-1

[CRGreathouse]

Originally Posted by science_man_88

my first thought based on the numberphile video would be the polynomial form $ x^{2n}-x^n-1 $ all numbers in the sequence for x=10,can't be divisible by 2,3,5, mod 7 it becomes 3²ⁿ-3ⁿ-1

I think Mini-Geek was talking about sieving much higher than 7.

__________

P.S. (S.B.): just wanted to add so that this topic was not interrupted, here, in place:
Only primes such that (5|p) = 1 can be factors; that is, in other words, only p>=11 that end with 1 or 9.

Similarly, for 10²ⁿ⁺¹-10ⁿ-1, only p :: (41|p) = 1 can be factors; that is, 23, 31, 37, 43, 59, 61, 73, 83, 103, 107, ...

[lavalamp]

Originally Posted by science_man_88

my first thought based on the numberphile video would be the polynomial form $ x^{2n}-x^n-1 $ all numbers in the sequence for x=10,can't be divisible by 2,3,5, mod 7 it becomes 3²ⁿ-3ⁿ-1

This is true, but it's also obvious without using modular arithmetic. 998999 is clearly not divisible by 2 or 5, and since all but one of the digits is divisible by 3, the number as a whole is not divisible by 3.