(Thanks to David Marcus who sent me the video I point to in point 4 of this post. Tip for young bloggers (if there are any) you can have a half-baked idea for a post and then someone sends you something OR you later have an idea to make it a full-baked idea for a post. That's what happened here. So keep track of your half-baked ideas.)
1) When I was 10 years old I wanted to find out how many seconds were in a century. I didn't have a calculator (I might not have known what they were). I spend a few hours doing it and I got AN ANSWER. Was it correct? I didn't account for leap years. Oh well.
(An astute reader pointed to a website that does the centuries-to-seconds conversion as well as many other conversions. It is here. If such was around when I was a kid, what affect would it have on my interest in math? Impossible to know.)
2) Fast forward to 2024: I wanted to find the longest sequence of composites all \( \le 1000\). One long sequence I found by hand is the following (I also include the least prime factor):
114-2, 115-5, 116-2, 117-3, 118-2, 119-7, 120-2, 121-11, 122-2, 123-3, 124-2 , 125-5, 126-2
length 13.
I wanted to find the answer WITHOUT using a computer program or looking at list of primes online (though I do allow a calculator just for division and multiplication).
Of more interest mathematically is trying to prove that there is no sequence of length 14. (If there is, then perhaps the attempt will lead us to it.)
Assume there was a sequence of consecutive composites \(\le 1000\) of length 14.
Map each one to the least prime that divides it.
At most 7 of them map to 2
At most 3 of them map to 3
At most 2 of them map to 5
At most 1 of them map to 7.
At most 1 maps to 11. (Can look at 11*p for all primes \(11\le p \le 89\) and see any of them are in a sequence of length 14.)
I'll stop here. This is getting tedious and might be wrong. So I asked Claude. It gave me a sequence of composites of length 19. Here it is (I include the least prime factor):3) The more general question: what is the complexity of finding the longest string of composites all \( \le n\) . This is actually many questions:
a) By hand: by which I mean only allowing multiplication and division and only of numbers \(\le n.\)
b) Theoretically. Use whatever fancy algorithms you want.
c) Theoretically but can assume some Number theory Conjectures that are widely believed. The Wikipedia page on prime gaps is here. (ADDED LATER- an astude commenter pointed out that we want LARGE gaps between primes, but the wikipedia article is about SHORT gaps between primes.)
d) Do a,b,c for the set version which is as follows: Given \(n\) and\( L\) determine if there a sequence of consecutive composites of length L that are all \(\le n\).
4) Does anyone else care about calculation-by-hand? Yes! There are people who want to compute\(\ pi\) to many places JUST BY HAND. Here is a video about them here. Spoiler alert: they did very well.
