Base 3
- The problem yesterday was unfair.
- I am sorry. In what way?
- You said Euclid had proven something he hadn’t!
- Did I? Oh well, something I get things mixed up.
- Is that your only excuse?!
- Have you heard of Imre Lakatos?
- I am not quite sure. Did he write ‘How to effect children’s affections in rural Albania’?
- No, he wrote ‘Proofs and refutations: the logic of mathematical discovery’.
- I was not even close. Was he Albanian by any chance?
- He was Hungarian. More importantly, he suggested textbooks should be written in heuristic style to show how problems are solved and mathematics created.
- So the textbooks should be less authoritarian?
- Exactly, if something is true or not is for the student to ponder, not for the textbook to impose.
- Sounds like an impractical approach.
- Anyway, here is a problem Ravi a reader in India posted in a comment in New York Times the other day.
- Is it inspired by Lakatos?
- Give me a prime number.
- How about 11?
- What is 11 in base 3?
- You want me to write 11 as the sum of some 9s, 3s, and 1s?
- OK.
- 11 = 1*9 + 0*3 + 2*1. So 11 is 102 in base 3.
- Add 11 and 102.
- 113.
- Which is prime!
- So it is.
- Find the biggest prime that when added to its base 3 representation is a prime.
- What if there are infinitely many of them?
- Hadn’t thought of that!
June 18th, 2010 at 5:04 am
89?? = 10022?
Now if you arbitrarily change 10022? to 10022?? and then arbitrarily add 89?? to it, you get 10111. This is prime.
There are 8 similar cases in the first 36 primes, mostly clustered at the top of the list. I have no reason to suspect this is anything more than random chance.
If I am wrong, I can’t wait to read why. Stranger things have been shown to be true.
June 18th, 2010 at 5:06 am
grrr…. base numbers didn’t show up. Here’s my post again in a readable format:
89 (base 10) = 10022 (base 3)
Now if you arbitrarily change 10022 (base 3) to 10022 (base 10) and then arbitrarily add 89 (base 10) to it, you get 10111. This is prime.
There are 8 similar cases in the first 36 primes, mostly clustered at the top of the list. I have no reason to suspect this is anything more than random chance.
If I am wrong, I can’t wait to read why. Stranger things have been shown to be true.
June 18th, 2010 at 7:59 am
“- Find the biggest prime that when added to its base 3 representation is a prime.”
Michael is in the lead with 89. Anyone better?
June 18th, 2010 at 10:49 pm
Go to http://www.wolframalpha.com/ and type ‘PrimeQ[1931]‘. The answer is True. In other words 1931 is a prime. One may also type ‘Primes between 1930 and 2010′ and it is on the list of primes returned.
Then type ’1931 to base 3′ and 2122112 is returned. Finally, type ‘PrimeQ[1931+2122112]‘ and True is returned.
In other words, 1931 is the biggest number so far that when added to its base 3 representation is a prime.
How did I find 1931? I used a spreadsheet with functions that converted to base 3 and told me if a number was prime or not. The functions I found, of course, on the Internet.
When I copied the rows a bit more down I found 6011 as a new record.