Teenager solves stubborn riddle about prime number look-alikes • Hacker News Board

Re: Teenager solves stubborn riddle about prime number look-alikes

#41

Post by phkahler » Thu, Oct 13, 2022, 7:32 PM UTC

renewiltord wrote: post #22

Summary: 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3. 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers. 3. Okay, so these numbers exist. How common are they? For primes we know they're common.…

But from the article:

>> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X.

And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he prove?

Re: Teenager solves stubborn riddle about prime number look-alikes

#42

Post by bumby » Thu, Oct 13, 2022, 7:32 PM UTC

MikePlacid wrote: post #25

BaseballPhysics wrote: post #17

Earlier quoted context omitted.

> I would rather guess that it is some genetic defect in the brain causing a person to prefer playing with abstract problems to booze, smoke and sexual gratification. But I doubt that having such a guess is allowed. Understandable given it's a comically elitist point of view. Fun fact: Richard Feynman experimented with both LSD and Ketamine, among other things. Shame, imagine how much he could have achieved if he had…

Elitist?? I doubt that any American family has a lower standard of living, than a Soviet math post-graduate student, single mother of two. We have no permanent beds only folding ones, I made my studies on a drawing board put over a sewing machine (do you know what sewing machine is for? It’s to repair your old clothing) our apartment was shared by two families, it has no hot water and water itself was de facto ration…

You might not know the wide variability in American standards of living. I had a SO who did not experience much luxury growing up. They had to run an extension cord from a charitable neighbor to have electricity to do their homework. It wasn't uncommon for them to sleep bundled up in winter clothes together in the living room because they had no heat.

I understand television may give a false impression of the American lifestyle, but there's a wide range of experience in a country approaching 400MM people.

Re: Teenager solves stubborn riddle about prime number look-alikes

#43

Post by ok_dad » Thu, Oct 13, 2022, 7:35 PM UTC

I knew a guy in high school that carried around a sub-compact notebook and one day in science class we were learning about how to factor quadratic equations (a review of old math we should know) and this guy was not paying attention at all, just typing away. The teacher asked him what he was doing that was so important that he couldn't listen, and to please come up and solve the problem.

This kid walked straight up to the board and explained how you can design a computer program to factor any polynomial equation string input to it, and in fact had implemented a polynomial equation factoring program while the teacher explained how to factor simple quadratic equations.

Since then, I don't feel bad if someone achieves more than me, because clearly there are some people out there that are born to solve certain classes of problems (maybe their brain structure is better for those, or something, who knows).

Re: Teenager solves stubborn riddle about prime number look-alikes

#44

Post by bibanez » Thu, Oct 13, 2022, 7:37 PM UTC

phkahler wrote: post #41

renewiltord wrote: post #22

Summary: 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3. 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers. 3. Okay, so these numbers exist. How common are they? For primes we know they're common.…

But from the article: >> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X. And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he…

I only read the abstract and the result is in the same spirit but doesn't say exactky between X and 2X. It's between some more complicated expressions (using logs like usual in number theory)

Re: Teenager solves stubborn riddle about prime number look-alikes

#45

Post by paulpauper » Thu, Oct 13, 2022, 7:39 PM UTC

363849473754 wrote: post #36

4gotunameagain wrote: post #19

Getting older, sometimes it can be so tough to accept the fact that people a fraction of your age achieve things you never will. Given the extreme connectivity of the present, we are also exposed to brilliant minds with incredible capabilities, making us (me at least) feel even more incapable.. I guess it is a lesson for humility. Good job Daniel, you show us !

Not to downplay any of Daniel’s accomplishment but sometimes it isn’t a “fair” comparison when others started younger with more resources. His father is a distinguished professor of mathematics and his mother is a professor of mathematics. When you have that sort of resources available at a young age and advanced training you’ll probably accomplish more sooner than someone of similar IQ without those resources who st…

There are thousands of mathematicians in the US. I am sure many have kids. How many of those kids do even a fraction of what Daniel did even when having every possible advantage? Today, young people have assess to more resources than ever, yet talent is one of those things that resists this trend of egalitarianism seem elsewhere. More resources means that the super-talented will pull way ahead of the untalented or only moderately talented.

Re: Teenager solves stubborn riddle about prime number look-alikes

#46

Post by 363849473754 » Thu, Oct 13, 2022, 7:43 PM UTC

paulpauper wrote: post #45

363849473754 wrote: post #36

Earlier quoted context omitted.

Not to downplay any of Daniel’s accomplishment but sometimes it isn’t a “fair” comparison when others started younger with more resources. His father is a distinguished professor of mathematics and his mother is a professor of mathematics. When you have that sort of resources available at a young age and advanced training you’ll probably accomplish more sooner than someone of similar IQ without those resources who st…

There are thousands of mathematicians in the US. I am sure many have kids. How many of those kids do even a fraction of what Daniel did even when having every possible advantage? Today, young people have assess to more resources than ever, yet talent is one of those things that resists this trend of egalitarianism seem elsewhere. More resources means that the super-talented will pull way ahead of the untalented or on…

To be honest I care more about the high iq kids interested in math who come from poor or working class families that either won’t be identified or will be identified but not much can be done for them given the lack of resources. Plus this doesn’t negate that someone of a similar IQ may accomplish less / seem less impressive at an early age due to such disadvantages.

Re: Teenager solves stubborn riddle about prime number look-alikes

#47

Post by renewiltord » Thu, Oct 13, 2022, 7:44 PM UTC

phkahler wrote: post #41

renewiltord wrote: post #22

Summary: 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3. 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers. 3. Okay, so these numbers exist. How common are they? For primes we know they're common.…

But from the article: >> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X. And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he…

Ah I didn't mean to mislead that it was x and 2x. Corrected to be slightly clearer.

The bound is easiest seen on the arxiv link above, in the abstract. The HN forum software doesn't do math very well.

Re: Teenager solves stubborn riddle about prime number look-alikes

#48

Post by c1ccccc1 » Thu, Oct 13, 2022, 7:45 PM UTC

BaseballPhysics wrote: post #17

MikePlacid wrote: post #15

Earlier quoted context omitted.

Dunno. My granddad was a mathematician and my mother was a mathematician (and my father was a mathematician and a computer scientist, but they divorced). Still the only “math immersion” I got was Perelman books, scattered around our apartment, like Physics for Entertainment and Algebra for Fun: https://www.amazon.com/s?i=stripbooks&rh=p_27%3AYakov+Perelm... They are both challenging and entertaining and very simple -…

> I would rather guess that it is some genetic defect in the brain causing a person to prefer playing with abstract problems to booze, smoke and sexual gratification. But I doubt that having such a guess is allowed. Understandable given it's a comically elitist point of view. Fun fact: Richard Feynman experimented with both LSD and Ketamine, among other things. Shame, imagine how much he could have achieved if he had…

Is that really true about Feynman? I thought he wrote in "Surely You're Joking" that he didn't take psychoactive drugs because he loved thinking and he "didn't want to mess up the machine".

Re: Teenager solves stubborn riddle about prime number look-alikes

#49

Post by renewiltord » Thu, Oct 13, 2022, 7:45 PM UTC

vorticalbox wrote: post #32

renewiltord wrote: post #22

Summary: 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3. 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers. 3. Okay, so these numbers exist. How common are they? For primes we know they're common.…

Thanks for this. I find articles like this super hard to read due to the mixing of the topic and all the "back ground". It does sometimes feel like it's only there to bulk out the article.

One can hope that future AI autosummarizers can be aware of our personal level of knowledge!

Re: Teenager solves stubborn riddle about prime number look-alikes

#50

Post by paulpauper » Thu, Oct 13, 2022, 7:47 PM UTC

munk-a wrote: post #38

4gotunameagain wrote: post #19

Getting older, sometimes it can be so tough to accept the fact that people a fraction of your age achieve things you never will. Given the extreme connectivity of the present, we are also exposed to brilliant minds with incredible capabilities, making us (me at least) feel even more incapable.. I guess it is a lesson for humility. Good job Daniel, you show us !

I celebrate people who achieve memorable milestones like this or build something that revolutionizes the world - but there are a plethora of different ways to live life. For me, as long as you're happy and you're putting more happiness out into the world than you're consuming then you're achieving a pretty damn good life. We're not all responsible for the entire world - as long as you're leaving your little corner of…

It's not enough to make the world better or be happier, a lot of people also want acclaim and recognition. Why do so many people apply to Ivy League schools when 50-100 ranking schools can also provide a good education? Status is necessarily scarce.